CF, what's your body language like when you're asking for a down? Are you bending over at the waist at all? Or standing up tall? Often, with dogs who are unsure of us, we need to make sure that WE project that we're in charge. If you're bending over even slightly, you're not conveying that. So watch your body language.
Also, it might help if you make a game out of it while you're in public around dogs. Play with him, so that you're doing fast sits and releases (or stands). Quickly reward the sits and release him. Over and over again. About 10-15 times. Fast fast fast. Then, when he's all caught up in the game, ask for a down. Probably, he'll plop into a down without even realizing it. Jackpot reward that. (Lots of itty bitty snacks all in a row.) Woo hoo! Make a big deal of it quickly and release him. Then start the game again. Do it a couple of times, then move away from that area, like "wow, that was a great game, wasn't it?" Don't stay there, like you were trying to sneak him into a down the whole time.
Eventually, you can work in more downs into your "fast sits" game, so it's 5 sits to a down. Then you can make it a fast downs game. The point is that it's a wonderful game. You're not leaving him in the down to feel vulnerable.
And of course, the more often and the more places you do this, the more comfortable he becomes lying down in public. (*At least* 15 places before a behavior is generalized). Then you can start with short duration downs (with lots of rewards during the down) and do that in your 15 places. Within about a month or two, he should feel better about lying down in public areas. But if a dog approaches him for the next several months, I'd give him a sit command and let him pop up. He doesn't need to "break" his down. You can anticipate and let him follow your command.
That will help him see you as a Safe leader, which is the whole point.