TL;DR: Not true
I'll go the easy route and steal from Snopes:
Claim: Donald Trump stated that Muslims should be made to wear identifying badges.
WHAT'S TRUE: Donald Trump was asked in an interview about whether Muslims should be subject to special scrutiny, a question he answered ambiguously.
... (I'm skipping the second half of the "true" sentence about databases, because it will be discussed later in the answer).
WHAT'S FALSE: Donald Trump asserted that Muslims should wear identifying badges
To elaborate, Snopes clarifies:
In the context of that interview, it's important to note that Trump's responses were non-committal. Furthermore, they were clearly in response to leading questions for which the actual phrasing wasn't even provided:
Precisely how such a question was presented to Trump was not elaborated upon in the printed text of the interview, nor was what his exact response (not "rul[ing] it out") entailed. Moreover, the portions involving quotes were so exceptionally vague ("do things that we never did before," "certain things will be done") and full of obfuscation, it was impossible to discern even vaguely what Trump referenced. (The mandate of badges for Muslims was quite a leap by any measure.)
While it appeared Trump fielded a question about enhanced surveillance for Muslims and mosques, in no reasonable interpretation of the material provided did he himself suggest that followers of Islam should wear Holocaust-like badges as in Nazi Germany. That assertion appeared to be one fronted by the interviewer, and not fairly attributable to Donald Trump. It's true that Trump espoused a position many would deem objectionable or offensive in the little he did say, but the controversy hinged largely on words he didn't appear to have said.
A little bit more organized, if more partisanly-worded, rebuttal of the claim with details of how it was spun up is here.
P.S. There was a second part of the claim, that was NOT part of this question; but was related and analyzed by Snopes as part of the same article.
... He then later affirmed that Muslims should be required to register in a database - Evaluated by Snopes as part of "TRUE" claim.
I actually disagree with Snopes in this "true" evaluation. This part of the claim is somewhat true but very misleading given context - no matter what interviewer's intent, Trump was NOT speaking of all Muslims, just immigrants. Which can of course be spun into something awfully sounding until one realizes that all legal immigrants to USA have been registered with INS/DHS, forever.
Note Trump's exact words around the clarifying discussion - they are almost certainly referring to immigration:
But right now, we have to have a border, we have to have strength, we have to have a wall, and we cannot let what’s happening to this country happen any longer.”