Why is it a custom in Baptist Churches to vote on new congregation members?

This practice is done in Baptist churches mostly as a practical matter, rather than a spiritual one. I'm speaking mostly from an Independent Fundamental Baptist perspective, but I believe that the same holds true for most Baptist denominations.

IFB, as the "Independent" part suggests, have a particular emphasis on individual church polity and the rejection of any mortal man or group of men to have any specific authority in the church, which Christ being the only head, and with pastors simply being under-shepherds. Ultimately, Christ runs the universal church, and each individual church, and they each follow as well as they can by His leading. The Bible basis for that would be verses such as:

1 Corinthians 3:11 - For other foundation can no man lay than that is laid, which is Jesus Christ.

Colossians 1:18 - And he is the head of the body, the church: who is the beginning, the firstborn from the dead; that in all things he might have the preeminence.

The leadership attempts to seek God directly for all matters, and tempers themselves with the counsel of those elders in the church who have walked with the Lord for a long time.

Unfortunately, there are some practical considerations. At least in the United States, to be able to be considered an official organization, and to hold property and other necessary considerations, a group has to have a certain number of members and bylaws. Those laws are crafted with an eye toward legal concerns. Rather than allowing a single person or group of people within the church to claim the whole legally recognized body, most choose to follow a democratic system. Most that I know of follow something like Robert’s Rules of Order. This allows them to make decisions that at least meet the basic legal requirements and seem fair without leaving power to one man.

When it comes to joining the church, the same method is applied. As far as we are concerned, what makes a person saved is the sanctifying work of the Holy Spirit in a person's life which makes them a new creature. It's a work done by God, and not one that is controlled by man. For a person to be a part of the actual church, this must have occurred. But we don't have any rational, objective method, at least not which would hold up in court, to demonstrate that this has occurred. Therefore, we settle the matter by vote.

We talk with the prospective members about their salvation. If they have not been saved by God, we encourage them to be so, and advise them that we can't knowingly allow somebody to join the secular organization when they are not part of the spiritual one.

Assuming they have been saved, there are typically three methods to accept that profession. One way is that they can be baptized in the church to make that public profession that they are a new creature and that they should begin to walk as such. A second method is that they can join "by letter", meaning that if they had already been baptized by a church of like faith, then that congregation can vouch for their consistency in walking in the faith and that they know God is working in their lives. Finally, a person may join by a "statement of faith" where they simply state that they have done these things, but without a letter. This is often done if the old congregation can't be contacted, or if there were some error there.

Once the pastor has done all that he can to confirm that the new members are in fact already part of God's church, then they present that fact to the congregation to make it an official membership and accepted by secular authorities. This would also provide an opportunity for anybody who had knowledge that a prospective member was not saved to be able to bring it up, but I've never known that to happen. In practice, it's just a formality. It gets the member on the books so that when the church applies for their status as an organization, they meet the membership requirement. These people also become voting members, and then have the ability to vote on other business matters, such as how the money is spent, what missionaries to support, and also on other members upon their joining and leaving.