What size feeder do I need for a basement electrical subpanel?
National Electrical Code says that for general lighting and receptacle circuits, you can use 3 volt-amperes per square foot to calculate the load. However, the square footage is calculated from the outside dimensions of the floor area. So when calculating the area, don't forget to include the wall thickness.

National Electrical Code 2014
Chapter 2 Wiring and Protection
Article 220 Branch-Circuit Feeder, and Service Calculations
220.82 Dwelling Unit.
(B) General Loads. The general calculated load shall be not less than 100 percent of the first 10 kVA plus 40 percent of the remainder of the following loads:
(1) 33 volt-amperes/m2 or 3 volt-amperes/ft2 for general lighting and general-use receptacles. The floor area for each floor shall be calculated from the outside dimensions of the dwelling unit. The calculated floor area shall not include open porches, garages, or unused or unfinished spaces not adaptable for future use.

Let's say you're going to be servicing a 1300 square foot area with the panel, and it sounds like you're only planning for 120 volt general lighting and receptacle circuits. The load would be calculated as follows:
3 volt-amperes * 1300 sq.ft. = 3900 VA
3900 VA / 120 Volts = 32.5 amperes
In this scenario, you'd be able to use a 40 ampere double pole breaker in the main panel, and 8 AWG copper or 6 AWG aluminum conductors to feed the panel. However, you may want to leave room for future expansion, so the minimum might not be what you're looking for.
You may in fact want to use a 60 ampere double pole breaker, and 6 AWG copper or 4 AWG aluminum conductors. Even though you only need half that capacity, the cost difference might be worth it if you want the option for future expansion.