If every person is guaranteed a basic income who is there left to work/be employed? I think only more ambitious people will work. What will motivate the rest of the people?
The level of the basic income would have to be set at a level that the majority of the population is still willing to work to get more money than the basic amount, so that required work is still done. (The robots are not going to be taking over all the jobs for some time). From the proposals I have seen, it would be near the poverty level in the developed countries.
At that level, I believe there have been enough experiments to validate that most people would want to work to earn more than the minimum. There is a lot of studies showing that many people want to contribute, rather than sit around.
The problem may be effect on tax system. This other question on menial tasks seems to be related to your question. If there is no increased taxation to match the increased transfers, inflation would be quite high, given the size of the program. (This is discussed in my answer on the linked question in more detail.)
Brian's answer surely hit a major part of the issue - the level of the UBI. Yet, there are other aspects of the problem.
- People like to work. There is a huge amount of sociological literature showing that people derive considerable non-pecuniary benefits from work, or alternative, that lack of work leads to several personal (and social) undesirables (for example, here, here, and here). As Voltaire said,
Work keeps at bay three great evils: boredom, vice, and need.
Thus, many people will like to work. This is the supply side of the labour market.
- Firms will still need workers. Provided capital and robots do not take over all jobs, firms will still require some workers. If the supply of labour across skill levels is not high enough, then either migration will occur (perhaps politically undesirable, but historically favoured by firms), or wages will go up. The latter will induce more people to work.
The key is, then, not only the level of UBI, but whether there will be enough jobs. Perhaps there will not be, at least not for the current working week. Maybe Keynes' prediction of a 15-hour working week might finally become true.