Obviously, each case is unique and everyone should consult with an attorney to get a legal opinion, but this is a very complex situation, and can have serious consequences not just for a work permit under DACA but also for future steps like DREAM Act, green card (permanent residence), and eventually citizenship. There is no doubt that identity theft is a federal crime and states have other related laws, and as far as immigrants are concerned any claim at any time to be an American citizen is a serious problem even if you are not prosecuted for it (prosecutions for identity theft alone are rare but when it happens in conjunction with another crime then prosecutors will add a charge).
Surprisingly, the US Supreme Court has ruled (in the case of Ignacio Flores-Figueroa) that an immigrant who simply uses a SS# (belonging to someone else that he or she purchased from a criminal or simply pulled out of thin air and depending on what it is, it may or may not belong to someone else) for the purpose of employment is not committing identity theft because he is not knowingly using that information belonging to someone else (and he is not stealing that person's name, address, date of birth, etc. only the SSN). So theoretically, if you used a number that belongs to no one, you have committed no crime. On the other hand, if you cooked up a number that turns out to belong to a real person, you still may not be in trouble as long as the intention was not to cause deliberate harm to this individual. In that sense those individuals who have been using social security numbers belonging to someone they know well (because it is a deliberate action even though the person may have given you implicit permission to use it, though, that itself is a crime), or it belongs to a real individual, may have more to lose than those who simply picked a random number or bought a number from a crook.
So what are the implications? It is indeed true that at every point in the immigration process, various agencies would want to know if you have ever used other numbers or claimed to be a US citizen (those illegal immigrants who simply check the US citizen box in an employment application in order not to submit any proof of their authorization to work are in deep trouble when it comes to legalizing their status) and the outcomes may depend on individual situations, ranging from outright rejection of applications to prosecution and deportation.
Chances are that most applicants for this process will be approved for a work permit without any hassles (it is not expected that the USCIS will refer cases of identity theft to prosecutors because DACA merely defers action on deportation for two years at a time but does not grant any legal status), but in many ways the process of enforcing these laws is totally broken. Many applicants can get away with no consequences, but depending on individual situations, it could also result in complications with approvals of legal status/permanent residence/naturalization. It is best to discuss this with a lawyer well versed in immigration and identity theft matters.
Should you apply for DACA if you are in a situation like this? The simple answer is that right now there is no guidance from USCIS how it will treat applicants who have used a fake social security number for employment. Even if the agency does not bother with prosecution, it may simply reject the application due to the crime committed. What may happen, though, is that at some point, USCIS will clarify this because many attorneys are already screaming about this. In addition, applicants with this problem who have already filed and are either approved or rejected will go online to provide updates on their cases. So if you want to be super-safe, in addition to working with an attorney to deal with this dilemma, you may wait and not file your application till there is more information available.