First, I should define my use of the terms. By open communion I mean opening the communion service (the Lord's supper, the bread and wine) to anyone who professes to be a Christian by faith in Christ, regardless of his or her status concerning baptism or church membership. As far as I have observed, there is no truly "open" or "unrestricted" communion; all churches restrict it in some way. It's just a matter of how much it is restricted. By open membership in Baptist churches, I mean opening the church's membership to persons who have not been baptized by immersion. This varies in actual practice, and, I think, may be demonstrated sometimes as even receiving some who have only infant baptism. Apparently Baptist Union of Great Britain churches practice that, as well as some members of American Baptist Churches in the USA. But I believe most would only apply this in the case of one who received sprinkling or pouring after affirming faith in Christ.
Second, it is my proposition that open communion leads to open membership. I do not propose that every case of open communion leads to open membership, but that the practice of open communion over a period of time will generally lead to open membership. I will post below some evidence of the relationship between the two. If others have such evidence, I encourage you to post it in the comment thread. Those who have some evidence to the contrary should post that as well.
THE CASE IN THE NORTH CAROLINA BAPTIST STATE CONVENTION
The North Carolina Baptist State Convention had a controversy that was caused by some churches receiving as members those who had not been baptized by immersion as believers, and the reaction of other churches to this. In 1972 the Convention established a committee to report on the problem. The committee asked the "differing" churches to identify themselves. Although over 20 churches were thought to be involved, in 1973 only twelve churches voluntarily identified themselves and eleven of them presented their case to the committee. It can be seen in the churches' defenses of their practices that they definitely had made the connection between open communion and open membership. This is documented in Perspectives in Religious Studies 1977 - Documents Concerning Baptism and Church Membership: A Controversy among North Carolina Baptists (c. 1977 by Association of Baptist Professors of Religion). The cover also carries the heading, Special Studies Series No. 1. It should be noted that all the churches did not have exactly the same practice, and that they did not necessarily acquire it from one another. The general nature of the policies was that "we...accept into membership baptized believers from other Christian groups, whatever the form of their baptism...(p. 8)." And from pages 38 and 39: "Another determining factor for some of these churches moving to their present position was that they considered a discrepancy in the custom of Baptist congregations practicing 'open communion' but 'closed membership'.
What to us was the inconsistency of recognizing the Christian standing of so many while not opening to them the doors of membership was further impressed upon us as we considered our practice concerning the Lord's Supper. For many years we had given explicit recognition to the membership of other Christians in Christ's Body by sharing the Lord's Table with them in worship. If to invite them to the Table was to affirm our belonging to each other in Christ, how could we continue to refuse them membership if and when they sought it?...If we believed, as we did, that the Lord would receive them at His Table, how could we not receive them into our fellowship especially since that fellowship was ours only because we recognized that we were his.
We extend to a Christian from another Christian body the right hand of fellowship as a brother in Christ and welcome him as we welcome any Christian to the Lord's Table.
There is another way we may possibly differ from our fellow Baptist churches. We believe that no one is qualified to participate in the Lord's Supper who has not been baptized. One must be "in Christ" before he may receive Christ into himself as a member of a body of faith. Many Baptist churches across this state, however, practice "open communion" and "closed baptism." This means, in effect, that such churches offer the Lord's Supper to people whom they do not recognize as having been baptized. That may be a not-insignificant heresy. In any event, we know of no church of any denomination, which follows this practice, other than some Baptist churches. It is a practice which we are unable to embrace.
There are only two ordinances of the Christian church - communion and baptism. Some years ago we realized that closed communion was a dividing rather than a unifying element among Christians. And so we took steps to invite all Christians to take holy communion with us. Our present policy regarding baptism tends also to divide rather than unify and the acceptance of this proposed change would tend to encourage unification.
On the day of her organization by the Mecklenburg Baptist Association the proposed Statement of Faith was amended to provide for "Open Communion," the inclusion of all Christians in the observance of the exceedingly profound Ordinance of the Lord's Supper. Across the years we struggled with conscience in the contradictory demand that practicing Christians desiring to transfer from another Christian Church be required to submit to a re-baptism by immersion in order to share our fellowship.It is interesting especially in the third church quotation that they held that baptism was a prerequisite to communion and thought this was the general belief of all churches except some Baptists. Of course, the discrepancy was resolved in their case by recognizing non-immersions as baptism.
THE BAPTIST UNION OF GREAT BRITAIN
The following is copied and pasted from the Baptist Union website. It would indicate that some BU churches will accept persons without further baptism beyond infant baptism. What if I were baptised as a child? Many people who turn to Christ have already been baptised as a small child. If this applies to you, be baptised again as a believer. Baptists believe that baptism without faith is not the baptism of the Bible. What if I have been confirmed? If at your confirmation service you meaningfully confirmed the promise made on your behalf by your godparents, then some Baptist churches will welcome you on your profession of faith.
The bold emphasis is mine to show the point I want all to notice. I have not found specific quotes to show the BU churches' connection of thought between open communion and open membership. The quote above is just to substantiate the claim I made about the Baptist Union in the first post. It appears that at the point in time the BU is consistent in its view of open communion and open membership.
Excerpts from the following article by Rowland Croucher discuss Open Membership in Australian Baptist Churches. I have given his categories of membership, as well as his "further reading," in which one may find interesting material on the subject. Croucher sees the open/closed communion/membership questions as being related.
(Paper presented to the School of Ministry, Whitley College Melbourne, July, 1992. A 2010 update may be found HERE.)
...The relationship of a person's faith-to-baptism-to-church membership is one of the most complex issues in contemporary Baptist faith and practice. Most Australian Baptists have resolved the closed/open communion issue (in favour of open communion), but only a minority of our churches have moved to an open membership position...The key question: 'Why is it possible to be accepted into the family of God but not into the family of a Baptist church?' Baptists have given many answers, which can roughly be summarized into six broad categories:  HARD CLOSED: Here members are only those baptized by someone with authority in one's own Baptist denomination. Many U.S. Southern Baptists, for example, will re-baptize other Baptists.  SOFT CLOSED: These churches will not re-baptize someone already immersed as a believer, unless the baptism took place in a sectarian group.  MODIFIED CLOSED: This - with the 'soft closed' position - is the stance of most Australian Baptist churches. Here a believer who is unbaptized, or was baptized as an infant, is given 'associate' status, and may vote on secondary matters in church meetings, and generally will not be eligible for the office of deacon or elder.  MODIFIED OPEN: In these churches only those who are baptized can be members, provided the individual regards their baptism - of whatever kind - as valid for them. This is the position of about 70-80 of our Australian churches.  PLURALIST OPEN: These churches (eg. in parts of the UK and in North India) go one step further and allow options for either infant or adult baptism, choosing sprinkling, effusion or immersion.  WIDER OPEN: This position allows the individual, in consultation and prayer within the community of faith, to reach a conclusion about baptism that is valid for them, but may be a full member of the church during this process. Further reading: T. Bergsten, 'Baptism and the Church', The Baptist Quarterly, Vol 18 nos. 3 & 4 (1959), pp. 125-131, 159-171. D. Bridge and D. Phypers, The Water that Divides, IVP, 1977. A. Gilmore, Baptism and Christian Unity, Valley Forge, PA: Judson Press, 1966, pp. 58-74. Thorwald Lorenzen, 'Baptism and Church Membership: Some Baptist Positions and their Ecumenical Implications', J. Ecumenical Studies, 18:4, Fall 1981, pp. 561 ff.AMERICAN BAPTIST CHURCHES IN THE USA
The following represents the membership views of one Arkansas church in the ABCUSA:
Membership Information Membership at Judson is open to baptized believers of all Christian denominations. Our baptism should unite, not divide us. When we baptize, we practice immersion. We welcome as Associate Members those who wish to continue membership in their home church. Our members come from many denominations and our communion table is open to all believers. A welcoming class is held for all who desire membership in Judson American Baptist Church.The church web site may be viewed here. [This baptism information was copied in January 2002, but is no longer available.]
This is from an ABCUSA church in Colorado:
Calvary has open membership. In other words, if you have been baptized (regardless of the form) you are received openly as a member of this congregation based on your Christian experience. If you have been baptized as an infant and would find it meaningful to be immersed as an adult, you are encouraged to do so. Please talk to one of the ministers. Pastor’s classes are available for all ages.Their web site is here. [This was copied in January 2002, but is no longer available.]