Can a married woman enter heaven if she bears no children? Is there biblical basis for the idea that if a married woman bears no children she won't go to heaven? Is there a biblical basis against the idea?
Yes she can

You're probably getting tripped up by this passage:


  1 Timothy 2:13-15 ESV For Adam was formed first, then Eve; and Adam was not deceived, but the woman was deceived and became a transgressor. Yet she will be saved through childbearing—if they continue in faith and love and holiness, with self-control.


Which, by itself and read without any context, does seem to imply that a woman needs to bear children to be "saved". But we should know better than to read passages without their surrounding context. Context is king ☺.

This idea that women need to bear children to be saved is, of course, totally contradictory to the rest of the new testament and any of Paul's other writings. Paul made great effort to emphasize the fact that it takes faith in Jesus Christ to make peace (i.e., get saved) with God.


  Romans 2:28 ESV For we hold that one is justified by faith apart from works of the law.
  Galatians 3:11 ESV Now it is evident that no one is justified before God by the law, for “The righteous shall live by faith.”


So then what does the Timothy passage get at? Well, it's not totally clear, and when it's not clear you need to be open to reaching multiple valid conclusions (like this essay does). A popular viewpoint (and the one I take) is that Paul was speaking specifically about Eve--that Eve should save herself and everyone by eventually giving birth to the Christ.