Contractions In Your Letters
Yes, Virginia, contractions are acceptable in writing letters. But, they must be used with discretion. Just as you shouldn't bullet everything on a page, you shouldn't make a contraction out of every possible word in a letter.
Yes, contractions do help the reader. Readers are used to hearing words in the contracted form. So, when readers see "would not," they turn it into "wouldn't." That means that using contractions speeds reading. Another benefit is that many readers miss the second word and take the exact opposite meaning. They read "would not" as "would." That doesn't happen when the word is "wouldn't."
Most people will agree that contractions are less formal than writing out both words. As in any other type of writing, it's important to focus on your reader. If you would speak more formally to someone, then you should probably write to them the same way. That may mean that in limited cases, contractions are not appropriate.
In a few cases, there is a difference in tone between the contracted form and the two word form--"can't" and "cannot" or "don't" and "do not."