For example a customer has three payment types.
Cash, Credit, or Check

Would it be best to store these three option in an Enum or a String[]?

If it is better to use an enum, would you store the string value or the int value in the database?

Wrong way round IMHO. Decide how you are storing the data (surrogate or natural key) and then build the app to suit that. Personally, I would use natural keys and cache this (likely unchanging) data at the client. Not bother with enums at all.

pootle flump ur codings are working excelent.

There is no such thing as a lookup table. Treat this data no different than any other data in your database. If your standard is to use natural keys, then use natural keys. If your standard is to use surrogate keys, then create a surrogate key for these values.
And if you not referring to surrogate keys, but actually mean true .net "enums", well the whole practice of dynamically creating surrogate database keys in the applicaiton layer is so lame that I don't really want to get into it.

what happens if there are additional types eg debit card
in my books it has to be a table, ENUMS are fine for the very limited occasion where every possible value in perpetuity is known at design time, but even then they can still fail if you say move the db to another language. the often quoted example are Male / Female using M or F as the Key. but that can fall down in these modern times with a move to non English, or the addition of other categories eg Trannies

I'd go along with the idea of not alwasy using surrogate values for the key, it can make the data a heck of a lot easier to read.

eg
£: Cash
A: Account
C: Credit Card
D: Debit Card
.......
rather than
1: Cash
2: Account
3: Credit Card
4: Debit Card

That's so unpolitically correct that I love it!

And remember, if you are using a surrogate key, you should nigh-on-always declare a unique constraint appropriately too!

Unpolitically correct and inaccurate. That is not another gender in any book. Do you mean transgender as opposed to transvestite?

pootle flump ur codings are working excelent.

I'll bow to your knowledge on the subject