My home church has a policy of asking non-believers not to participate in communion, but never asks if a believer has yet been baptized.
I have visited or attended several other churches where communion is withheld from anyone who has not been baptized, regardless of whether or not they profess a faith in Christ. What is the Biblical basis for setting the bar at baptism as opposed to a profession of faith?
The biblical basis for restricting communion to the baptized is indeed 1 Cor 11:27-28. But, as @Richard pointed out, the interpretation of that passage isn't standardized.
Take a look, if you will, at the context of those two verses (NRSV). Paul is addressing a controversy where it seems some of the Corinthians were coming together to stuff themselves, and excluding others. That too is the consumption of the Lord's Supper in a disrespectful manner.
At any rate, at the supper that Thursday night long ago in the upper room, we have no clear words from Jesus about only serving the baptized. Were all the ones present with Jesus baptized? Who knows? He did serve all of them.
There's an early church document that isn't in the canon called the Didache XII Apostolos (the Teaching of the Twelve Apostles). See here for example. That document spells out the requirement for baptism before partaking of Communion.
"Now no one should either eat or drink from your thanksgiving meal, but those who have been baptized into the Lord's name. For about this also the Lord said, "Do not give what is holy to the dogs."
It is very likely that the reservation of Communion to the baptized is a matter of church tradition as much as it is of Scripture.
It seems that regulations and opinions on this are varied. It depends (very strongly) on the denomination and what they believe.
I'll put this here for later reference:
27 So then, whoever eats the bread or drinks the cup of the Lord in an unworthy manner will be guilty of sinning against the body and blood of the Lord. 28 Everyone ought to examine themselves before they eat of the bread and drink from the cup.
One belief is that the communion should be something solely for people who are saved, committed Christians.
As The Rev. Gerald Schrankler said in this thread:
If we invite non-baptized, non-Christians, to communion we degrade it from a celebration of our redemption by Christ's death and resurrection to a fellowship snack.
The background idea behind this is that Christians who are not baptized are not saved. Since they're not saved, they should not be able to partake of the communion and are in violation of 1 Corinthians 11:27.
Another Protestant View
John Wesley, however, believed that communion was a "means of grace" (source). Therefore, it shouldn't be withheld from anyone--believers or non-believers.
The idea here is that if someone wants to come to communion, we should not prevent them. It may very well be the means that leads to their salvation! Clearly that's not something we would want to withhold.
Catholics actually allow others to take part in the Holy Communion, with tight restrictions.
Guidelines for the Reception of Communion
For Catholics
...A person who is conscious of grave sin is not to receive the Body and Blood of the Lord without prior sacramental confession except for a grave reason where there is no opportunity for confession....
For Other Christians
...Members of the Orthodox Churches, the Assyrian Church of the East, and the Polish National Catholic Church are urged to respect the discipline of their own Churches. According to Roman Catholic discipline, the Code of Canon Law does not object to the reception of communion by Christians of these Churches.
For Non-Christians
...While we cannot admit them to Holy Communion, we ask them to offer their prayers for the peace and the unity of the human family.
Ultimately, it seems that there are very widely dispersed opinions on this matter. The biblical basis as you asked, is 1 Corinthians 11:27-28. However, the interpretation of that verse has lead to varied regulations.
