The question is a little vague. The first thing to consider is that the traditional DB-9 or the older but still present DB-25 connectors have contacts in them which can sustain up to 1 Amp. I don't think that is what you are asking however.
There are no pins in a serial port connector that are intended for delivering power. There is a signal ground. Each of the other pins is an RS-232 signal. Those signals may be capable of 10-20 ma. Each serial connector will be different and there are no standards that I am aware of that deal with current sourcing.
The inputs have a reasonable impedance, of 3K to 7K, and should not require significant current to drive.