I'm going to offer this as two answers: one assuming you are talking about liquid sourdough starter, and one assuming you are talking about a liquid sourdough batter.
Liquid sourdough starter is simply regular sourdough starter with a lot of water (typically 5x the amount of water vs flour). The differences that the water brings to the starter are mostly a matter of flavor, as the different starters will still ferment and rise like normal. A liquid starter not rising would make sense as there isn't a lot of gluten structure to trap the air, so the appearance of a healthy liquid starter would resemble that of a ginger bug or lacto-fermented vegetables - it will just make a lot of bubbles that get released when agitated.
Using liquid starter is the same as using virtually any other kind of starter. Simply take the greater water content into account when calculating the ratios of your dough and otherwise use it like normal.
Liquid sourdough batter isn't necessarily a bad thing (after all, sourdough pancakes are delicious), but it's highly unusual for making bread - even extremely high hydration breads still need to be dough before they can become bread.
The reason your dough is liquid has nothing to do with the fermentation. Dough becomes dough because of the flour, and in this case, it's because of the gluten in the flour. The point of the mixing and kneading step of forming the dough is to develop the gluten so that your dough has structure, so if your dough is liquid, you need more flour. Otherwise, when it enters the rising stage, the "dough" will not rise much if at all simply because, like with liquid starter, there isn't a proper gluten structure to trap the gasses being produced in fermentation.
If time is a concern, you can certainly add commercial baking yeast, and that will definitely kickstart the rising activity. The downside, though, is that the resulting bread will be noticeably less sourdough-y and taste more like just normal bread. Some people add commercial yeast to their sourdough precisely because of this - they like sourdough but don't want it to be too strong. Whether or not this is an acceptable compromise is ultimately up to you.
Now if your goal was to bulk-ferment a liquid batter before you add more flour to turn it into a dough, that is certainly an approach that would work (and would probably result in an awesomely soft and tasty sourdough), and this is all a moot point as you would just add the flour in the morning. At that stage, though, I would say that the sourdough starter has had plenty of time to replicate and spread throughout the dough, gobbling up most of the readily available food and nutrients, so I'm not convinced that adding commercial yeast at this stage would have much of a practical effect.
Other possibilities for kickstarting the yeast activity:
- Add yeast food. This can be as simple as sugar, but you could also add potato starch (or even instant mashed potatoes).
- Add more starter. If your dough needs more yeast, the most ideal solution would be to add the yeast that you started with.
- Move the dough to a warmer environment. Yeast is a living organism, and as such, it doesn't work as efficiently in the cold. (For best results, keep your dough in a 95-105°F/35-40°C environment. You can go as high as 115°F/45°C, but as yeast starts to die around 120°F/45°C, getting too close to the line risks imperfections in your proofing environment causing parts of it to teeter over the edge.)