Do you mean unemployment benefits? Like the check payments while you look for work?
If is that, NO you are not elegible for the folowing reasons:
1. Your asylum is not fully granted. So you dont have status yet.
2. I believe you need to be previously employed in order to receive these benefits AND you hadto be fired or something like that... not just "I want to look for a better job".
ONCE YOU HAVE YOUR ASYLUM GRANTED, you can try to apply for welfare (food stamps and cash aid) I believe, if you qualify you can get it for 6 months or something like that. BUT if you have been working and making "enough" for yourself and your dependents (based on your county income per person guidelines) you will not qualify for this.
Good Luck,