That will depend upon where you're sending them to!
A few crossword competitions might use online forms, where you fill in the blanks on your screen via your keyboard and then (after entering your contact details) you simply click 'Send'. (However I've never actually come across such a site).
Some competitions might allow you scan your completed paper solution and then either upload it via a web form or email it. However the instructions for each competition are likely to be different. (e.g. some competitions might insist that you scan your completed crosswords into pdf format, whereas other will accept jpegs).
To the best of my knowledge though, the vast majority of crossword competitions still don't accept online entries. You have to post them.
If you've got any particular crossword competitions in mind, please let us know which they are, so we can advise you as to whether online entries are accepted (and the relevant methods).