I have owned my HE washer & dryer for over 2 years now and I LOVE them. I have the LG WM3875HWCA (washer) and DLEX3875W (dryer). As far as maintenance goes, after a load in the washer, I do leave the door wide open as well as the detergent drawer open temporarily -- usually only while I have clothes drying. The model I have has a built-in magnet that allows you to close the door without latching it so air can circulate but you won't have to have it wide open. After my last wet load I take a dry microfiber cloth to dry off the inside glass (only because I don't want water spots). I also quickly dry the rubber gasket which literally takes 10 seconds. Every 2 months I use an Affresh tablet to clean the machine on the tub clean cycle. In addition to that, I do one bleach load every week (bed sheets and bath towels), but I don't really have to do anything special to keep it clean.
Now, there ARE downsides to HE washers.
The washing cycles take way longer -- they use less water, therefore need to have longer cycles for the clothes to get clean. The cycle I use most frequently takes an hour for normal clothes (cold water). If you do the Allergiene cycle or Sanitary cycle, like for bedding or pet blankets, you're looking at over 1 1/2 hours. It uses steam.
Note that drying time is WAY less, though. The clothes are less damp from the washing cycle so they dry faster. My dryer has 2 sensors in it that detect if the items are dry yet, and will shut off once they're dry so you don't waste electricity over drying clothes. I love it.
My whites do not come out as white with an HE washer because there is no soak cycle. My whites are still white, but not quite as white. I don't mind. The rest of the clothes are noticeably cleaner, and I use less detergent. Of course I use the HE detergent, but you can use much less than what the bottle says.
We noticed a HUGE drop in our water and energy consumption right away, and the old units we had weren't even 10 years old I don't think (they came with the house when we bought it).
Oh, one nice thing, at least with the model I have, you don't have to rebalance a load, the washer is self-balancing and will adjust the drum on its own! I used to get so mad when I would do a load of towels or bedding and several hours later I'd remember I had laundry to finish only to find out the stupid washer shut off because it was imbalanced!
I highly recommend the exact models I have. I don't know what your price range is, but they are so worth it. LG has very few repairs.
And, a few more highlights about my machines, when you put your laundry in the washer, it will weigh it and tell you how large your load is so you add the correct amount of detergent. Mine literally holds twice as much as my previous "king-size capacity" washer. It also has an option for wash-optimizer, where if you may have put too much or too little detergent in, it will adjust accordingly and either use less water or more water so the balance of detergent-to-water is correct. The dryer also tells you if there is a blockage in the ductwork/vent. All I can say is, these things are amazing!