I've made several posts on the main Drupal site that have been downvoted with no explanation. Is there any way to see who is doing that?
No, there is no way to do that. There is a longer and relevant discussion on the topic here: A call to provide explanations for a down-voted question
After taking a look at the two questions you have with a -1 score, my guess is that Module to allow multiple paths for a single node? was considered "to obvious", since the functionality is already in core. Showing that you had attempted to figure this out on your own first might have made a difference.
This one on the other hand https://drupal.stackexchange.com/questions/28575/images-uploaded-using-media-gallery-w-plupload-do-not-attach-to-node very much looks like a bug, belongs in the issue queue, and is as such off-topic.
SE developers are the only ones who can see that data, not even diamond moderators have access to it. I didn't downvote your question about Plupload but I did vote to close it as off topic (for the reason @Letharion mentioned).
The important thing to remember is don't take it personally!
Voting up/down/to close a question is merely about quality control for the site; every question needs to stand on its own and be potentially useful to future visitors. If others in the community don't think a particular question holds up to those criteria, then downvoting/voting to close are the only methods (besides commenting) that any of us can use to voice that opinion.
I do agree it can be frustrating to receive downvotes without a comment though; as far as I've seen in the SE network it's very much encouraged to leave a comment so that the question can be improved upon.
There isn't a way to see who down-voted you, as votes on posts are anonymous by design.
It is not even an information that a normal user would need to know, as there is a mechanism to auto-detect votes irregularities (both up-votes, and down-votes) and correct those votes by removing them. Moderators can see suspiciously high levels of activity between users. This means that moderators are able to detect cases of cross-voting, where you vote a user's posts, and that user votes for your posts.
In your case, there isn't anything suspicious, as you got 4 down-votes, since April 25, 2011; of those down-votes, one is the automatic down-vote given to questions closed as off-topic. You also got 10 points removed because users who up-voted your questions, and then removed their up-votes.
To add to what Clive said, leaving comments about what is wrong with the post is recommended, but down-voters are not forced to leave a comment for the simple fact voting is anonymous by design. If they were forced to leave a comment by the software, you would have a chance to guess who down-voted, except in the case there are more comments from users who didn't down-vote, but that probably happens rarely, on small Stack Exchange sites.
The general reason of a down-vote is given in the tooltip for the voting button.