Agrichemicals sounds like they make for an uninviting swim, but where do the pollutants come from you ask? Just uphill of the springs - and an area you'll need to pass through if you descend to the springs - is a giant farm which also features a man made reflective pool with a rock garden. As you descend from the road you'll snake your way through the fairly extensive farm which undoubtedly drains off downhill and into the very waters you seek to enjoy. The farm isn't even the only one around, but it's certainly the most obvious. The owners' website suggests they are environmentally friendly, but being ethically minded doesn't mean totally natural farming.
website dedicated to the area, and charge everyone 30NT per person to get access to the lake. The charge seems conveniently left off their website last time I looked, however.
A final note on the lake. There are a ton of spots in Yangmingshan that look similar to this, and they're free, so I strongly suggest you save your money. Frankly, it's a tourist trap. You show up to the lake, an hour plus ride away from Taipei - more if you take the bus - and then are unexpectedly prompted to pay money to see what is essentially a fallow rice paddy.
Back to the hot spring discussion. You'll need to either walk across a set of rocks and sandbags which traverse a scaling hot river from the uphill entrance, or you'll need to wade across the cooler waters of the downhill entrance with Teevas or your shoes off. From uphill, you use the sandbags to skirt a fence - designed to keep people out - and from downhill you'll pass a sign, and a fence, threatening you with a 15,000NT fine for being inside. There are a lot more reports of people being fined for being at the spring than there used to be.
On my last trip to the spring there were about 60 scooters and 15 cars parked at the Bayan bus stop. The spring was swamped, to say the least, and it wasn't peaceful at all. Weekdays are definitely the right time to go for crowd control, but unless you have a scooter, I'd suggest you use the time you would spend taking bus 1717 from Taipei to Bayan to instead head out to better wild hot springs than this one which you can also find online. Also, you won't get fined for being at the other springs.
Fun trees are all over this area!
The basic feeling about the true experience at these "wild" springs is something like this. Only without the syringes.
A final rant. Considering how pretty some of the hot springs are in Taiwan - I dare say more beautiful than this "wild' one - and with public pools for less than 100NT, I can't see why you'd want to go to the one forbidden spring in Taipei except to coddle an obsessive need for going "off the beaten path". Except this spring is now so ON the beaten path they're enforcing the no entry rules. If you're in search of the untouched, it exists in Taiwan, just not at Bayan.
The logistics. If you're really dedicated to hitting these springs despite my trying to warn you off, you'll find your way there as follows. Take Bus 1717 from Taipei and get off at the Bayan Hot Spring Hotel 八煙會館 stop. You'll then follow a road slightly downhill to where a big red sign saying "do not enter" starts the trail. If you drive, you'll turn off at 2甲 marker 5.1, with the same road and signage. You'll walk along a river, slowly winding uphill towards a big plume of mist. Be prepared to slog through the river at times since the bridge which used to span it was destroyed to keep people out. Finally, bring some food and some water, nose plugs if you hate sulfur smells, and a towel. Good luck!