Will your 89c51 reset? Depends on quite a few factors.

You mention the relays are switching solenoids. Are they DC or AC solenoids? There are a number of ways to solve the problem - with DC solenoids you can use diodes like you do with the relays or a MOV (metal oxide varistor) with DC or AC solenoids.

Other issues are when the load is turned on, does it cause to mains voltage to drop for a short time? Large motors are a cause of this, if the mains drops too much, then the voltage your 5v regulator sees may be too low and your supervisor chip will reset your micro.
The solution to this is to have sufficient headroom above 5v so that a input voltage drop will still allow the regulator to supply 5v and/or a suitably sized input capacitor. Your earlier posts suggested you are using a 6v rail for the relays - how are you getting the regulated 5volts? A 7805 regulator will not be too happy with only 6 volts on the input! Again, how will it cope with a drop on the mains input voltage?

In answer to your earlier query - adding diodes on the output of the ULN2003 will make little difference - to main issue is that there is at least a diode in the circuit (in the ULN2003 or elsewhere)!