IMO? Possibly, but I don't think so.
What creates upward or downward pressure on wages? Supply and demand, obviously. For any given job, the amount and conditions of available workers who are able to do the work will determine the price of that labor. For highly specialized medical positions, for instance, there are only a small number of doctors with the training and certification to perform that work (in fact, medical schools adjust their enrollment in proportion to projected demand for doctors, ensuring that the labor pool remains quite close to the job pool, and thus that doctors' salaries remain high). In which case, the doctors' expenses set the bar for salaries.
A call center, OTOH, only requires basic communication skills and a day or two of training. The pool of workers willing and able to do that work is much, much larger than the number of jobs available. So the employer can offer a low wage, and get a mixed bag of both lower-quality and higher-quality applicants. They can then hire only those that are higher-quality. They are now getting more value out of that employee than the low wage probably reflects, because the same would have been paid to a low-quality applicant if they had been hired (this is also why they don't need to worry about retention; workers are easy enough to replace).
In short, wages are not determined by worker quality, but by worker availability.
How will a BI affect this? It depends on how big the BI effect is, in several ways. The first is getting people to quit their current full-time jobs and opt for less complicated part-time work. I don't think firms will be willing to convert full-time positions to part-time positions for skilled employees. I think the options for a skilled employee will be remain where you are unaffected by BI, or quit and either 1) try to find unskilled part-time work that you're happy with, or 2) become an entrepreneur/artist/student/otherwise leave the "employer-employee" world. So how many people will stay where they are, how many people will move down the jobs ladder, and how many people will get off it entirely?
The other thing to consider is how many people already working multiple part-time, unskilled jobs will cut back? This, I believe, is where the effect will be largest. Very few will choose to work 80-90 hours a week across 3 different part-time jobs anymore. Most will scale back to working 1, maybe 2 part-time jobs.
So those who quit a full-time job and seek part-time jobs will raise the supply of unskilled labor, which would tend to lower wages, while those who cut back within the part-time labor force will decrease the supply of unskilled labor, which would tend to increase wages. There will also be some shuffling with the people who leave the labor force to pursue art/studies/raise children, etc., which will probably also put upward pressure on wages. The question is: will the number of recently "descended" workers looking for new part-time work exceed the number of workers cutting back their hours? If the answer is "yes," there will be, on the whole, more workers competing for the same jobs, which will likely decrease wages. If the answer is "no," there will be, on the whole, fewer workers competing for the same jobs, which will likely increase wages.
My estimate is that there are many more poor-ish people who will cut back to 30-60 hours/wk instead of 80-90 than there are middle-class people who will quit one job and try to get a part-time one. So I'm optimistic that, ceteris parabus, unskilled, part-time wages will increase. I don't think it'll be a significant increase, though.
How big of an effect will the increased negotiation power be? Not that big, I don't think. Again, because of the availability of other workers. The only way for that negotiating power to matter is if a very large number of workers assert it, and that is simply not going to happen without coordination. Unions are what increase worker negotiation power, not BI. Others say that because there is a viable alternative and less pressure to say "yes" in order to eat, workers will be able to hold out. Again, I say only with union coordination. BI certainly makes unions more palatable and effective, so that negotiating power might materialize, but only when a BI is tied to stronger unions.