Short answer : No.
It's not the executive branch's job to issue fines : it's the legislative branche which pass bills, and the Department of Justice issue the fines.
One could make the case that the administration - the president himself in fact - can issue Executive Orders that pass as laws, but it still can't issue the fines themselves and the laws are already in place anyway. If their are still businesses that hire illegals, it's because these laws, like a lot of laws, are hard to enforce in an absolute way. People are also mugged and murdered even though these acts are also illegal, after all.
Given the body of your answer, I understand that you may not ask about the administration itself issuing fines, but about what they could do to make these fines being issued. The answer is not simple :
- They can do nothing and let the DOJ do its job. It's already illegal, so employers can be prosecuted on that ground and be issued fines. i wrestled a bear once was kind enough to bring materials that list these fines : the legal documents are here and here and a layman description of the law can be found here. Thank you for your precision.
- They can work on a bill to reform the existing laws and/or increase the budget allocated to investigating the hiring of aliens. It would imply a complete analysis of how it currently works and using it to find ways to improve the current legislation and/or the way it is enforced, for example by removing possible loopholes.
Since your question also address the fact that Donald Trump said a lot of things about reducing/suppressing the presence/hiring of illegal immigrants in the USA and that your subquestion is Why wouldn't he do it if it was possible, I'll give you my opinion on this, which is that he surely don't care that much about that issue, at least.
Most likely he even likes it that way because it can be used as an argument to gain support : He hammered his wall story during his whole campaign even though it's a dumb idea because it is nothing more than a catchphrase to win the support of those who don't like immigration. It's easy to understand and it makes some sense as long as you don't think too much about it, nothing more.
As a side note, given that Donald Trump never dissociated his personal interests and his job as POTUS, I am also that he wouldn't put much effort in increasing the enforcement of these laws simply because he is likely to also benefit from the work of aliens.
Not having absolute elements regarding that point, I'll just leave it as food for thoughts, but he makes/made most of his money from the loans he secured with the buildings he built and has a history of not respecting agreements, including not paying contractors. I'd say he has little to no insentive in removing aliens from working given how a large proportion of them work in construction. Please bear in mind that this is an analysis, and that I write it only because the OP's second question can't be answered by references to facts.
I rewrote my answer because some points seemed to be ambiguous, leading to comments about things that weren't addressed in my answer or the question.