Your subject line says "Delhi airport (BOM)...". In fact the code for Delhi airport is DEL.

Your question is slightly more subtle than you may realize. The answer depends on which domestic flight you are taking, because a small number AI domestic flights are joint domestic-international flights. If taking those flights (which leave from the international side at DEL) you don't pass immigration or Customs in DEL; you do it at the final destination. If that is true, of course you have to go to and stay in the international departure area. You cannot leave the airport.

If you are not taking one of those special flights, you collect your checked luggage at DEL and pass immigration and Customs. You can then check your luggage again at the nearby transfer desk, and after that you should be able to leave the airport. Someone will correct me if in fact you are not allowed to leave once you check luggage---if that is so, just leave your luggage at the left luggage office for a fee, and check it in after you return.

Please post your domestic flight number and someone will tell you what type of flight it is.