A colleague in one of the physical sciences years and years ago claimed she stopped reading a manuscript after the first two or three typos, particularly if they appeared early on. The reason she gave was that her time, offered freely, was limited and she expected a certain level of professionalism exercised by the authors. Seems a bit harsh, but we all end up having to find ways to prioritize our time.

Based on this question, your written English is better than that in some papers I've reviewed.

The scholary periodicals are usually called "journals" not "magazines" :)

@Aruralreader There are grammar mistakes and there is sloppiness. I have reviewed papers which were full of grammar mistakes, yet were written with care, had a logical flow, and were easy to follow. And I have reviewed (and refused to review) papers where the author obviously didn't even read through the text once before submitting. In my experience, papers with bad English more often turn out to be the latter kind. A limited command of English is no excuse for sloppiness. And if the writing is extremely sloppy, what are the chances that the research work wasn't sloppy as well?

A warning word to the OP: The lack of question mark after your last sentence and the mistagging of this question ("graduate admissions") are exactly the kinds of things that make the impression of sloppiness (assuming such mistakes appear not once, but many times throughout your paper).

@Szabolcs I've seen some with grammar that was consistently wrong in a few ways - but perfectly consistent, and the vocabulary spot on. A native speaker of the same language as the authors (Russian) explained how they got there.

You should find a friend/mentor who is proficient in English and can comment on your manuscript prior to submission

Since nobody mentioned it yet, I'll do it: Please use the spell checker of your word processor. Back in my active days, I received lots of papers with terrible typos that would have been found by a run-of-the-mill spell checker. I tried my best to still judge those papers purely by their academic merits, but such an obvious lack of respect for the reader's time will introduce an (unconscious) bias against your paper. Yes, I know that your question was about grammar rather than spelling, but at least getting the spelling right already puts you ahead of a lot of other authors.

@Heinzi: Unfortunately, spell-checkers, particularly the real-time sort, probably introduce as many errors as they fix. Particularly when you do scientific writing, where many technical terms are seen as misspellings of more common words, and "corrected". I'd suggest a spell checker that works as a separate post-processing step.

@jamesqf: Good point, I didn't think of that. In my field, we almost exclusively use LaTeX, so spell-checking is always a manual post-processing step, never "autocorrect".

@Heinzi: Same here. But sometimes I run into software, like web browsers, that drive me crazy trying to autocorrect things (and drawing squiggly red lines under them) until I figure out how to turn off their spell check.

It is not very expensive to simply hire someone to review your paper for grammatical issues. Plenty of places exist online for this...

@Dúthomhas It can be very expensive, especially if you are a student and you have to pay it out of pocket.

Define "very expensive". If you are publishing a paper as part of your final PhD research, I would consider it part of the cost to polish it. I'm in the USA; a couple hundred bucks is less than the cost of a textbook. Even if you just hire someone from the English Language program at your uni for some reasonably comparable amount it is justified for the end text representing your work. IMHO.

@jamesqf: Some will allow adding new entries to the dictionary on the fly. The false positives can very quickly be reduced this way. It works very well in Firefox, and it doesn't autocorrect. It can also be turned off on the fly (for a lot of text it can become quite CPU intensive). For instance, while I wrote this comment, I added "autocorrect" to the dictionary (and to my list).

Focusing on my highball estimate for the USA is missing the point and your experience does not meet with the reality of textbook costs in the USA (which is still missing the point). Calling something ridiculous because you think it is does not help. Pay someone who knows English well something reasonable to get the job done. Polish at a critical juncture that will be reviewed by people you'd like to impress in the future is worth a little expense, just like paying for the costs of uni, no?

@Dúthomhas I think we have very different ideas about how much it is reasonable to pay for education :) (note that I am European, although I did my phd in the US). But regardless, it might be worth it, but it really should not be mandatory.

I think we are circling here. All I said is that you can pay someone for a totally non-mandatory English language review for the non-English speaker/writer, and that it is my recommendation for an important piece of paper. I would personally find 50 USD reasonable online, but would not put something a bit larger as unreasonable.

Other than cost, why can you not use a translator or some other kind of collaborator?