I am going to be controlling a relay with a reed switch. The behavior I am looking for is to have the relay switch on immediately (or as quickly as possible) when the reed switch is closed but when the reed switch is opened again, a few moments goes by before the relay is shut off... Just in case there is interference in the magnetic field of the switch. So if the reed switch were to turn off and on again for a second or so, the transistor would remain saturated.

However I am unsure of exactly how to get a capacitor to act this way. Most things I find on google talk about ramping up the voltage slowely, rather than trying to sustain it and having it drop slowly.

 

It should work, just put a capacitor below your switch and ground. (Must stay left of the base resistor). If it doesn't stay on long enough, you need either (or both)...
A) bigger capacitor
B) more gain and less base current on the transistor

I would use a 100uF capacitor and a 100k ohm pot for the base resistor (with a 1k ohm in series for safety). Then you can adjust delay as needed. Assuming you have a darlington with a out Hfe equal 1000.

Note: connect 100k pot as a variable resistor, not as a voltage divider.

 

I am going to be controlling a relay with a reed switch. The behavior I am looking for is to have the relay switch on immediately (or as quickly as possible) when the reed switch is closed but when the reed switch is opened again, a few moments goes by before the relay is shut off... Just in case there is interference in the magnetic field of the switch. So if the reed switch were to turn off and on again for a second or so, the transistor would remain saturated.

However I am unsure of exactly how to get a capacitor to act this way. Most things I find on google talk about ramping up the voltage slowely, rather than trying to sustain it and having it drop slowly.

If you have at least 6V headroom, you can switch the relay coil with a MOSFET - the gate leakage will take practically forever to discharge the capacitor - so you'll need to add a bleed resistor to leak the charge away.