For cryogenic service I agre with unclesyd: 304 has better properties than 316, but the difference is so small, that I think that the main reason which makes 304 to be used for cryogenic service is the money.
For cryogenic service (typical LNG or LN2) corrosion is not a problem (usually) so both of them are OK. If you have an important corrosion factor, or if you think the difference in money is not a problem in your case I would use 316, if not I would use 304.
And.. what about 304 vs 304L (or 316 vs 316L)??
I know the difference in carbon percentage, with the corresponding problem for welding (not so big actually) but, do you think one of them is better for cryogenic service??
Personally, if there are not other reasons or data in the process data sheet I use to select SA-240 304, and it works perfectly, but which are your opinions?
In cryogenic service pressure is not usually very high (for example in a LNG plant the highest could be 20barg in the recondenser) and so thickness are not very very big, so welding is not extremely difficult (I wouldn't use 304 for a reactor with a 200mm wall!).