No Stupid Questions

IANAL, and lawsuits almost always end up being very fact-intensive, which means that the specifics of the case often make the difference. So it'll depend. But broadly speaking, if there isn't a specific law -- eg ADA -- that specifically assigns liability, then the most typical claim someone would try to make is a theory of negligence. That is, failure of the laundromat to behave with a reasonable degree of care.

In the absence of signage or disclaimers or waivers (like in some amusement park rides), the jury will have to assess whether this laundromat's environment suggested some heightened sense of security (eg security cameras, even fake ones) or that management implied or leaned into marketing that made it sound like clothes wouldn't be stolen there. But a typical coin-op laundromat has people going in and out at all times of day, so it's not reasonable to think it's akin to Fort Knox, even without a sign indicating that management disclaims liability for clothes theft.

As for posting that sign, it won't change the general lack of liability on the laundromat in a case where someone snatches clothing. But the equation is different if, say, a patron asked a staff member to watch their laundry for 5 minutes as they make a phone call, and that staff member agreed but then went out for a smoke, resulting in an opportunistic thief stealing the $80 bras from the dryer. Here, the laundromat would carry liability, because although they don't normally watch the clothes, they agreed to do it this once and did it so badly that the clothes were stolen. That's negligence, despite the sign.

That said, posting a warning sign is generally encouraged, since a core principle of liability is that avoidance of harms is always going to be preferable than litigating after they've already happened. So if the sign causes patrons to stay near their clothes in the machine, then some amount of theft has been outright avoided. For this reason, courts seldom will punish a business for having an overzealous sign, unless the sign itself is materially false or the sign itself causes a hazard (eg a loose "Gusty Winds" highway warning sign that falls over in a light breeze, injuring a middle school student).

But to muddy the waters some more, another core principle of liability is that liability should fall upon the person whose behavior if changed will prevent future harms. For stolen clothes, it's quite clear that the thief should be liable for the value of the stolen bras. If a court instead holds the laundromat liable, then that creates a perverse incentive where rather than spending money on more/better washers, the laundromat must spend that money on cameras and private security, raising the cost of the laundry machines. In additional to absolving civil liability on the thief. All for something which would be more cheaply solved by patrons just watching their laundry, or perhaps installing hasps on the machines so patrons can bring their own locks.

On the flip side, denying liability means the patron has lost the value of their clothes. Perhaps they now have to spend more on "clothes insurance", which only serves to benefit an insurance company rather than affording more bras. Adjudicating liability -- in any legal system -- is a thankless job and there are never perfect answers to the delicate balancing act. Life is messy, and even the best civil tribunals struggle to make sense in all of the turbulent circumstances.

TL;DR: it depends