Many denominations practice closed communion. One traditional rationale for "fencing the table" is 1 Cor 11:29, glossed in terms of the real presence (although that's a dubious interpretation).
Here's another argument for closed communion.
As baptists we're not denying that paedobaptists have a right to their own perspective, we are simply maintaining the integrity of our own convictions. Our consciences will not permit us to welcome into membership and communion those who have not obeyed Jesus at the point of baptism.
This is the whole reason there are Baptist churches at all. This is why baptists don't commune with Presbyterians, though it doesn't close down the possibility of cooperation in gospel efforts that are wider than local church ministry (such as T4G and TGC). If this issue were not big enough to divide over, to deny membership over, then why did the baptists ever separate from the presbyterians?
One question this raises is whether Jesus practiced closed communion:
14 And when the hour came, he reclined at table, and the apostles with him. 15 And he said to them, “I have earnestly desired to eat this Passover with you before I suffer. 16 For I tell you I will not eat it[b] until it is fulfilled in the kingdom of God.” 17 And he took a cup, and when he had given thanks he said, “Take this, and divide it among yourselves. 18 For I tell you that from now on I will not drink of the fruit of the vine until the kingdom of God comes.” 19 And he took bread, and when he had given thanks, he broke it and gave it to them, saying, “This is my body, which is given for you. Do this in remembrance of me.” 20 And likewise the cup after they had eaten, saying, “This cup that is poured out for you is the new covenant in my blood. 21 But behold, the hand of him who betrays me is with me on the table. 22 For the Son of Man goes as it has been determined, but woe to that man by whom he is betrayed!” 23 And they began to question one another, which of them it could be who was going to do this. (Lk 22:14-23)
Now before the Feast of the Passover, when Jesus knew that his hour had come to depart out of this world to the Father, having loved his own who were in the world, he loved them to the end. 2 During supper, when the devil had already put it into the heart of Judas Iscariot, Simon's son, to betray him, 3 Jesus, knowing that the Father had given all things into his hands, and that he had come from God and was going back to God, 4 rose from supper.
18 I am not speaking of all of you; I know whom I have chosen. But the Scripture will be fulfilled, ‘He who ate my bread has lifted his heel against me.’ 19 I am telling you this now, before it takes place, that when it does take place you may believe that I am he. 20 Truly, truly, I say to you, whoever receives the one I send receives me, and whoever receives me receives the one who sent me.”
21 After saying these things, Jesus was troubled in his spirit, and testified, “Truly, truly, I say to you, one of you will betray me.” 22 The disciples looked at one another, uncertain of whom he spoke. 23 One of his disciples, whom Jesus loved, was reclining at table at Jesus' side,[e] 24 so Simon Peter motioned to him to ask Jesus of whom he was speaking. 25 So that disciple, leaning back against Jesus, said to him, “Lord, who is it?” 26 Jesus answered, “It is he to whom I will give this morsel of bread when I have dipped it.” So when he had dipped the morsel, he gave it to Judas, the son of Simon Iscariot. 27 Then after he had taken the morsel, Satan entered into him. Jesus said to him, “What you are going to do, do quickly.” 28 Now no one at the table knew why he said this to him. 29 Some thought that, because Judas had the moneybag, Jesus was telling him, “Buy what we need for the feast,” or that he should give something to the poor. 30 So, after receiving the morsel of bread, he immediately went out. And it was night. Jn 13:2-4,18-30.
The Last Supper is the paradigmatic Lord's Supper. The exemplar of the Eucharist. Yet Jesus administers the "sacrament" to Judas, even though Judas is a closet unbeliever–something known to Jesus. On the face of it, Jesus is practicing open communion.