Use the regular expression symbol \\W to match non-word characters, using + to indicate one or more in a row, along with gregexpr to find all matches in a string. Words are the number of word separators plus 1.
lengths(gregexpr("\\W+", str1)) + 1
This will fail with blank strings at the beginning or end of the character vector, when a “word” doesn’t satisfy \\W‘s notion of non-word (one could work with other regular expressions, \\S+, [[:alpha:]], etc., but there will always be edge cases with a regex approach), etc. It is likely more efficient than strsplit solutions, which will allocate memory for each word. Regular expressions are described in ?regex.
Update As noted in the comments and in a different answer by @Andri the approach fails with (zero) and one-word strings, and with trailing punctuation
str1 = c("", "x", "x y", "x y!" , "x y! z")
lengths(gregexpr("[A-z]\\W+", str1)) + 1L
# [1] 2 2 2 3 3
Many of the other answers also fail in these or similar (e.g., multiple spaces) cases. I think my answer’s caveat about ‘notion of one word’ in the original answer covers problems with punctuation (solution: choose a different regular expression, e.g., [[:space:]]+), but the zero and one word cases are a problem; @Andri’s solution fails to distinguish between zero and one words. So taking a ‘positive’ approach to finding words one might
sapply(gregexpr("[[:alpha:]]+", str1), function(x) sum(x > 0))
Leading to
sapply(gregexpr("[[:alpha:]]+", str1), function(x) sum(x > 0))
# [1] 0 1 2 2 3
Again the regular expression might be refined for different notions of ‘word’.
I like the use of gregexpr() because it’s memory efficient. An alternative using strsplit() (like @user813966, but with a regular expression to delimit words) and making use of the original notion of delimiting words is
lengths(strsplit(str1, "\\W+"))
# [1] 0 1 2 2 3
This needs to allocate new memory for each word that is created, and for the intermediate list-of-words. This could be relatively expensive when the data is ‘big’, but probably it’s effective and understandable for most purposes.