Sure, you could handwash everything "private" at home and save the outerwear for the laundromat. Personally, I haven't found handwashing to be nearly as effective as a machine wash, so this option doesn't appeal to me.
So assuming that you "have" to take everything to the laundromat, there are steps you can take to maximize your privacy.
- Pre-sort your clothing if you plan to have multiple washes, such as colors, whites, and delicates. If you plan to shove everything in the biggest washer to save a dollar or two, this step is unnecessary.
- Put your underwear and other "private" items at the bottom of the pile. This minimizes the surprise evacuees to the floor when you're putting your clothing into the washer. When all the private items are grouped together, you can get them into the washer faster and with less likelihood of them falling to the floor.
- Use an opaque hamper or bag to transport the clothing.
- If you don't have an opaque hamper, you can place other clothing or linens around the sides to shield your private items from public view.
- Place linens (sheets, towels, blankets, etc) on the top of the hamper if the top is open to the public.
- When transferring your clothes from the washer to the dryer, find a linen or large piece of clothing to place all the clothing inside, wrap it around the clothing, and then push it to the dryer. It's going to take more time and be more noticeable if you try to sort out your private items into a separate pile.
- If some of your private items can't go into the dryer, remove them from the washer and place them separately back inside your opaque hamper. If your hamper isn't opaque, it is probably fine on the bottom of the hamper. It depends on what your hamper is like.
- After they are dry, don't fold or hang your clothes at the laundromat. To be honest, just shove them back in the hamper, go straight home, and deal with your clothing there. If you're worried about wrinkles, not many will happen if you're close to home. And if they still happen, you can use a hairdryer to warm the clothes back up when they're on the hanger. (There are also clothing de-wrinklers available in stores.)
It's not a perfect solution, but it can make the laundromat trip more bearable.