I saw this in theatrical release when it came out, and just recently on home video. As others have commented here, seeing it now can clarify some of what changed in the assumptions of filmmaking during the 90s. This film isn't apologetic about taking its subjects fairly seriously, even while not getting too solemn or "taking *itself* too seriously" in the bad sense. What I'm trying to say is that it isn't flip, and doesn't feel that it has to elevate cool to the ultimate virtue.
If my code words there haven't made it clear, what I was objecting to was how Tarentino has influenced attitudes. But there's an entirely different kind of crummy moviemaking that actually came to mind as a contrast to Henry and June while I was watching this, and that was Moulin Rouge. Please excuse me for using one movie as a club to bash another, but while watching H & J I really felt this connection and contrast.
Even though the 30s (H&J) was a quite different era from the 1900s (Moulin), they're both distant enough from us to raise similar questions about how to handle that sort of milieu. And they have in common being set among artists and aspirants. "Moulin Rouge" presents Erik Satie and Henri de Toulouse-Lautrec as characters on the fringe of the story, tossed in for spice perhaps, but gives us not a hint about what Satie's music sounds like, nor a glimpse at any Toulouse-Lautrec canvas (though I credit the production design with getting something of his feel into how the film looks.) We are told the hero is an aspiring artist but have no idea what his work is like, what he thinks, what the artistic times were like -- except a vague profession of "all for love".
"Henry & June" makes pretty clear what Miller's and Nin's aims were in their writing, and even some of the flavor. It also fills in the artistic surroundings, with for instance some of Brassai's photographs as well as using him as a minor character, and a clip from Le Chien Andalou. The score (with the filmmakers realizing the not everything is of-the-moment, and the avant garde of the 30s would still be "processing" the avant garde of the 20s, teens, and before) gives us snippets of Le Sacre du Printemps and some of Satie's piano pieces. (It was hearing the Satie that triggered the comparison for me, since -- as above -- I've been using the absence of his music from Moulin Rouge as an indicator of the shallowness of its approach to its setting.)
The makers of Henry & June figured out that the best way to show us exteriors for Paris of the 30s is to selectively show us Paris of today. This would have worked also for Moulin Rouge, better than the gratuitous and grating special effects. True, some landmarks of the (18)90s and 1900s are gone, but that doesn't call for wholesale resort to animation.