I work at my university as a mentor for undergraduates applying to medical schools and medical students applying to internships and residencies. I also review manuscripts submitted by physicians/scientists at the start of their careers to the peer-reviewed journal I edit. In all these areas, accurate spelling, correct grammar, and even proper punctuation greatly influence how seriously I--and doubtless others in similar positions--consider the submitted material. What you write is a proxy for who you are, and careless and sloppy writing reflects strongly on the impression the acceptance committee or reviewers will have of you.
But, not to worry, there are a myriad of helpful and effective writing guides available. All offer some version of Strunk and White's classic advice, as summarized by Sword, in their famous book, The Elements of Style, first published in 1918 and continually revised:
Always use clear, precise language, even when expressing complex ideas; engage your reader's attention through examples, illustrations, and anecdotes; avoid opaque jargon; vary your vocabulary, sentence length, and frames of reference; favor active verbs and concrete nouns; write with conviction, passion, and verve.Another option is Helen Sword's recently published book, Stylish Academic Writing (2012). It is short, highly readable, entertaining, and a great help for those who wish to get rid of bad writing habits and communicate clearly, effectively, and with style.
If these two writing guides are not to your liking, many others are available. The most important step is to pick one guide (or two), read, learn, and write! For as Ray Bradbury once said, "You fail only if you stop writing." So don't stop, but do it well.