Yes you/we should. I am "almost a newcomer". Also, I am an amateur with almost no formal training at a college but I have some training at a college and some of my own self-studied.
And, although this is a site for professional mathematicians even some amateurs can surprise you with their questions and ideas, so, I think that we should award those who really show an effort and are doing their best to help community grow.
There is some injustice here done, and it is easily seen by some closings of some questions that are not of research-level by opinions of some of you here, although researchers sometimes are not able to answer them. I am talking from my own experience because I have and I had had such questions.
First, innovative spirit should be awarded, and there should be no or almost no envy toward some that show creative spirit (and I do not see myself as one of them) but I feel that some questions of mine are not awarded enough, either in the sense of responses or in the sense of upvotes.
It seems that amateurism is seen as something distant and strange, but are you sure that some amateurs cannot succeed in knowing all that you know?
This starts to look as some promotion of amateurs on the site for professionals so there is a feeling to stop talking about that relationship(s) here and to move on onto newcomers.
I am of the opinion that the more is someone endeavouring to know something by asking something brilliant inside well-studied fields or succeeds in creating/finding some new concept or new problem, that then she/he must be appropriately awarded, be it newcomer or no, but if a newcomer then she/he should be even more awarded, so to witness care and encouraging of the community here.
But if some newcomer asks something too trivial or not so appropriate for this site then she/he should be very politely leaded to other options,for example, to ask for help on MSE (if the question is too trivial), or she/he should receive careful explanation in the comments of why that question of hers/his is not appropriate for this site. If some of you have "power" to just click the button "close" to some question(s) without explaining in the comments why the question is not for this site then , if you ask me, some kind of injustice is done.
I would never personally (or at least I hope I would not) close some question just because it does not reveal something "deep" about distribution of zeros of Riemann zeta, or it does not go far enough into the Hodge conjecture, because, some questions that do not appear to be for this site, can, after studying of them, appear to be very fruitful and inspirational, and can reveal something that was not seen at the first glance.
Treat newcomers as you would like to be treated if you were a newcomer.
Explain as precisely as possible why something is not for this site, or why it seems that it is not, and avoid closing because your personal interpretation is such that you just, because of some reasons of personal nature, think that the question is not so appropriate as you would want it to be.
Sorry if I hurted someone, or if I missed the point, just wanted to say a few words.