I guess I still would. The clear case is when one can answer a question. The regular practice of picking up a question that is accessible, answering it, and communicating the answer can be very beneficial. Even I am learning from it a lot and I'm by no means new to "random problem solving". So, if a student (or anybody in general) can answer questions on MO, I would definitely encourage them to participate. Just warn them that it may become addictive, so they should balance it with their regular duties. I have also never seen any "unwelcoming behavior" with respect to people who post answers, at least, as long as purely mathematical discussions are concerned.
The issue arises when somebody's contribution is reduced to asking endless questions of the type "I cannot figure out this place in the textbook, can you help me?". There are questions of this type that are still eligible for MO, but many of them are on the borderline and would be better redirected to MSE or just to the student adviser. So, I would suggest that in such cases the students view MO as the last resort, not as a help hotline.
Any questions about student research (i.e., something that is not readily found in the textbooks or, if you prefer, something that falls under the "missing-lemma" tag) have always been welcome and I try to answer them and to protect them from closing as much as I can even if they look a bit "homeworkish". I would encourage such questions any time. Some of them are now being asked on MSE, which is a pity, because they usually just get flushed by the stream there unless somebody attaches a bounty.
As to "MO being less friendly to newcomers", my feelings are mixed. On the one hand, we introduced a lot of friendly perks like the automated reminder to (try to) be nice whenever responding to someone who posts for the first time. On the other hand, there still seems to be a strong movement to eradicate everything that is not "research-looking" when judging from pure appearance (though I would say it was much stronger a year or two ago). I mean such things as, for instance, voting to close for "the lack of motivation". My own stance is that any clearly posed mathematical problem that I cannot solve in under 10 minutes should pass and the OP has no obligation to explain in detail where it came from and what it is related to. It especially applies if the OP is a student: quite a few of them have enough trouble with merely understanding and stating what exactly they need.
The MO may be not exactly what it was 10 years ago (all is changing, all is flowing, and some part of the discrepancy in perception may come from the fact that we ourselves are not exactly what we were 10 years ago), but it still looks like a nice and interesting place to be at to me. As soon as the student is ready for a mathematical discussion, he or she is welcome to enter. That may happen at any "stage of PhD" and at any age. I wouldn't be surprised to learn if some participants of MSE who ask there questions that might be suitable for MO as well are, in fact, high school teenagers or undergraduates. If in doubt, just try MSE first, but if you see that your questions merely earn you reputation there but are left unanswered, switch to MO.
Those are just my two cents, but I'm an old-timer. It would be really interesting to hear what the newcomers have to say, especially the young ones.