Trump hasn't even taken office yet so no blame can really be tied to him just yet. He's said quite a few silly things so far, but hasn't yet done any real damage. Most of the hysteria is being generated by the 24 cable news channels. I can't turn on CNN without another Trump-o-phobia headline on the screen.
To be fair NATO/Europe have been having issues during the Obama administration. Brexit is just a symptom of a long running problem in the EU regarding national sovereignty and bail outs of poor nations (greece/spain). Obama has struggled finding partners in the middle east to provide support to fight terrorism, but even NATO allies like Turkey have refused to even let the US use their airspace.
I personally think that Trump is picking his battles. He can fight a Mexico trade/immigration battle and get an easy win. Fighting Russia over Syria has no winners. As for Assad, it's hard to see how a person responsible for the deaths of over 300K of his own people could ever really hold on to power with or without foreign support. The US has experience in supporting an unpopular foreign leader (south Vietnam) and it didn't go that well.
I think in the long run, Trump will be more of a 'domestic' president. Someone that really spends more time focusing on domestic issues rather than foreign ones. Hillary I think would have been the opposite. In some ways this may be better for foreign countries, because it means that we are more likely to stay out of their business.