Why is it a custom in Baptist Churches to vote on new congregation members? My friend recently became a member of the local Baptist church. When she and her mother wanted to join, they went to the preacher and he told them there would be a vote among the congregation members to determine whether they would be accepted into the church. She and her mother had to stand before all the members, and with a show of hands, the congregation agreed to their joining the church. 

It struck me as odd because it never occurred to me that there was an acceptance process for the Baptist church. Is this true of all Baptist churches? And what is the scriptural basis for this process?
I'll start out with the last part of your question.  There is no explicit Biblical basis for this.  Voting on membership into the Church is an outcome of several Baptist distinctives, the primary being the autonomy of the local congregation.

From the Baptist Distinctives site's article on Church Governance


  Polity is how an organization, such as a church, functions—the
  policies that guide matters such as governance, decision making,
  structure and leadership. Baptists differ from most Christian
  denominations in matters of polity. The difference especially is
  evident in how congregations of Christians are governed.
  
  One major difference between Baptists and many other denominations is
  that no person or group outside of a Baptist congregation is to have
  any authority over the church in regard to beliefs and religious
  practices. Furthermore, all of the members within the church
  fellowship are to have equal voice in the governance of the church.
  
  Baptist church governance often is termed “democratic.” In a sense it
  is. In a democracy, all of the people have equal voices in decision
  making. No individual or group of persons is in control. Such is to be
  the case in a Baptist church. One way that democratic governance is
  practiced is that each member of the church has the right to vote on
  matters at church business meetings.
  
  To many non-Baptists, and even to some Baptists, this seems to be a
  strange way for a church to function. Putting the governance of a
  church in the hands of persons who have no special training, education
  or calling appears to be foolish. Why would Baptists dare to function
  in this fashion?


The article goes on to list several reasons for governing a Church this way.

To clarify, when the Church is voting on membership, they are not voting on whether a person is saved, or even whether they believe a person is saved. We're  merely voting on whether to allow the person in question to become a voting member of the Church.  

From the article on Regenerate Church membership


  What Are Other Factors Related to Membership in a Baptist Church?
  Churches differ on whether persons should be baptized and admitted
  into membership immediately upon their profession of faith or whether
  there should be a delay. Some baptize persons very soon after they
  have made a profession of faith. Others require persons, especially
  children, to go through a process of counseling before being baptized.
  
  Some churches require all persons seeking membership to be counseled
  concerning their experience of salvation and commitment to church
  membership. Some also expect persons to attend a class for new
  members. Many do neither.
  
  Baptist congregations vote on a person’s request for membership. The
  congregation is not voting on whether the person is saved or not. That
  is a matter between the individual and God. Rather, the church members
  are participating in Baptist congregational governance under the
  lordship of Christ.


Other relevant disctinctives that bear on voting on membership include Jesus as the only Lord (not a Pope or any governing authority), the Priesthood of all believers, 
the belief that all Church members should be regenerate (born again) Christians.