I talked about this with some other people, and I think the simple tweak to reduce XP for encounters by half and more, and using the variant rule to have one short rest per night and long rests as a week off in a secure town or stronghold should go quite a long way to improve things in this regard.
Casting create or destroy water as a 1st level spell produces enough water for 5 characters even in hot environments. That's one 1st level spell per day to completely negate the water needs of an average party. But if you have to cross a desert for a whole week, and don't regain spells until you reach the safety of the other side, that starts to seriously bite into a druids spells total for the desert crossing. A 3rd level druid would have to use all spell slots to provide water for five people for eight days. Characters won't likely die from dehydration in the desert if they don't bring water, but that threat would hover over them any time they get into a fight and the druid is consider using a spell to help with the immediate threat.
Cure wounds looks similar. It becomes a considerably greater investment any time you use it, if you have to ration your spells for a whole week or even two.
Of course it wouldn't emulate BECMI, but those two tweaks alone should result in a significant change in pace and tone, and they are about the most common variant rules I've ever seen discussed or mentioned.
As others mentioned, the exploration mechanics in 5th edition can barely be called a system. But I think in that case, you can pretty much plug in the old BECMI system as it is without even having to do much in the way of modifications. Use wandering monster checks and tables, use reaction rolls, apply the Morale mechanic. And I guess make the return of a treasure to a town a milestone that awards XP.