Applying for asylum is important for many people fleeing from danger and persecution in their home countries. However, it can take a long time for an asylum application to be approved, and it can be stressful waiting for a decision, especially without knowing what your legal status is. Maintaining legal status in the US is crucial for later applying for lawful permanent residence, which can be done a year after an approved asylum application.
Maintaining legal status includes not overstaying a visitor visa, so submitting an asylum application as soon as possible is incredibly important. This does carry a risk, however; while many consider a pending asylum as an indicator of legal status within the US, exceptions can and do happen, and a particular officer may decide not to consider the pending asylum application as proof of legal status. In this situation, the pending asylum is considered only as temporary permission to stay in the US, not as an actual status in itself.
In the precedent case of Kavafoglu v. Nielsen, the court determined that a family whose B-2 status expired but who had filed an application for asylum within the validity of said status was not in lawful status, and could not obtain a green card.
Every situation is different, and this may not affect you, but it is better to be aware of the possibility rather than to be caught off-guard.