ocool_15:
Your configuration will not work as you hope, both the transistor and capacitor are not correctly configured for this application.
True, the transistor could be used in either common emmitter (gained voltage) or common collector (unity voltage gain). I would prefer to use a unity gain circuit with the 12V relay because it would make RC calculations simpler, but if you calculate the values making use of transistor gain and minimum base voltage that could be made to work as well. A FET even might be more desirable due to it's high gate impedance that would not interfere as drastically with the RC time constant, but it all depends on the values you select.
Note for the calculations of R and C that RC = T and that is only for one time unit (~63% of full supply voltage applied to cap).
A simple switch like this would work fine IMHO, but remember that if you choose to use the transistor in gained configuration, that the relay will stay on (therefore your amps will stay on) for about the same amount of time after you ground the remote line. If values were calculated for the opposite transistor configuration the turn-off delay would be much shorter than the turn-on delay which might be another desirable feature.
I have modified your posted schematic and attached it below.
-Matt