From this excellent site:
One of the most frequently asked questions about colons is whether we should begin an independent clause that comes after a colon with a capital letter. If the independent clause coming after the colon is a formal quote, begin that quoted language with a capital letter.
Whitehead had this to say about writing style: “Style is the ultimate morality of mind.”
If the explanatory statement coming after a colon consists of more than one sentence, begin the independent clause immediately after the colon with a capital letter:
There were two reasons for a drop in attendance at NBA games this season: First, there was no superstar to take the place of Michael Jordan. Second, fans were disillusioned about the misbehavior of several prominent players.
If the introductory phrase preceding the colon is very brief and the clause following the colon represents the real business of the sentence, begin the clause after the colon with a capital letter:
Remember: Many of the prominent families of this New England state were slaveholders prior to 1850.
If the function of the introductory clause is simply to introduce, and the function of the second clause (following the colon) is to express a rule, begin that second clause with a capital:
Let us not forget this point: Appositive phrases have an entirely different function than participial phrases and must not be regarded as dangling modifiers.
There is some disagreement among writing reference manuals about when you should capitalize an independent clause following a colon. Most of the manuals advise that when you have more than one sentence in your explanation or when your sentence(s) is a formal quotation, a capital is a good idea. The NYPL Writer’s Guide urges consistency within a document; the Chicago Manual of Style says you may begin an independent clause with a lowercase letter unless it’s one of those two things (a quotation or more than one sentence). The APA Publication Manual is the most extreme: it advises us to always capitalize an independent clause following a colon. The advice given above is consistent with the Gregg Reference Manual.