The most obvious literal translation would have been de Russie avec amour. This translation, however, makes no sense in French when taken in isolation and is not idiomatic.

The title from Russia with love is supposed to mimick a common greeting from an intimate letter or a postcard, so the translator has tried to find an equivalent in French. Bons baisers de XXX is a common phrase used in such contexts in French (even though a bit old-fashioned nowadays).

Baiser only has a vulgar meaning when used as a verb (and even then, it has kept the meaning of “to kiss” in some fixed phrases with body parts or objects, eg baiser la main, baiser l’anneau).
The noun baiser as in this title does not carry this vulgar tone, so it is not shocking here. There is no ambiguity it refers to a kiss. You can find it in many other titles (“Le Baiser”, a song by Alain Souchon, “Premiers Baisers”, a French sitcom for young teenagers – imagine the outrage if this title was seen as vulgar !)