You can download the number of votes per day from the Stack Exchange Data Explorer with this query:
SELECT CreationDate, VoteTypeId, COUNT(*) FROM Votes
WHERE (VoteTypeId = 2 OR VoteTypeId = 3)
GROUP BY CreationDate, VoteTypeId
ORDER BY CreationDate, VoteTypeId
(Note that VoteTypeId 2 is an upvote, VoteTypeId 3 is a downvote). If you download the information into a CSV, you can graph it using (for example) R:
library(ggplot2)
votes = read.csv("QueryResults.csv")
votes$CreationDate = as.Date(votes$CreationDate)
votes$Type = factor(c("Up", "Down")[votes$VoteTypeId - 1])
ggplot(votes, aes(CreationDate, X, col=Type, group=Type)) + geom_line()

Of course both the number of upvotes and downvotes has been increasing with the number of users, and varies greatly by the day of the week, but there is no cyclical trend by season immediately visible.
Now, since there's so much noise within a week, this noise may be hiding a real trend in terms of "% of downvotes", which is what you're actually interested in. So let's look at % of votes each day that were downvotes. We'll also use local regression (LOESS) to smooth over time:
library(data.table)
votes = as.data.table(votes)
ratios = votes[, list(ratio=X[2]/sum(X)), by=CreationDate]
print(ggplot(ratios, aes(CreationDate, ratio)) +
geom_line() + stat_smooth(method="loess") +
ylab("% of Downvotes"))

Now, I still don't really see a seasonal trend (though I might note that the recent peak in downvotes was in the summer). But this does illuminate an interesting unrelated trend: that the % of downvotes decreased until about 2011 and then has been steadily increasing since. I suppose there are many possible hypotheses that could explain that!