As far as I know, on the 737-200 there is no single big overriding reason to start one engine prior the other. The company I fly for used to have a SOP for starting #2 first, but dropped it, with no explanation about a year ago.
Some minor reasons could be attributed to the arrival procedures. Where the SOP is to taxi in one engine, usually #2 is shut down because the belly baggage/cargo doors are on the right. This is the side the rampies should be hustling over to, and also where the baggage carts are parked/unloaded. If #2 is wound down by the time you roll up to the gate, there is less chance of FODing that expensive engine with a rampie
, bag, or trash.
Since you taxi in on #1, if you start #2 first, and taxi out only on #2 (if practical) then the engine run times will be more equal, and more importantly if running several short haul sectors while tankering fuel, the fuel load will stay more balanced.