If we envision an eventual time with mostly-automation or full automation, there will be a transitioning period. Say that there are jobs for 50% and the rest have basic income: how will those two classes look? The people that are working will probably want to be compensated fairly well, since they are dedicating their time and sometimes taking on heavy responsibilities (like doctors). But how will the folks on basic income handle this new income-divide? Will they be content with their hobbies, non-profit work etc; or will they be jealous of the people with (I'm guessing) a substantially higher income and all their status symbols and trappings? Will this mean a much, much more competitive labour market, starting from early school and continuing into university and eventually a more demanding wokplace? Will the "jobless" be seen as a new low-class, being looked down on not simply for having low-skill jobs while at the same time working hard, but truly being people with no responsibilities (like I said, non-profit work and such might be a possibility)?
In short; will human nature be able to gracefully handle the transitioning period?
First of all, the idea of basic income is that it is universal: there won't be a divide between those who have basic income but not work, and those who have work but not basic income. Everybody will have basic income; the divide will merely be between those who have work and those who don't.
The solution this divide is to eliminate the minimum wage, which acts as a significant hurdle to creating employment. In a basic-income world, there is no more moral argument for minimum wage to exist, because basic sustenance no longer comes from labour. Everybody will have sufficient food and shelter whether they work or not, so the exploitation of labour is no longer possible, and the safeguard of minimum wage is no longer needed. In a basic income world, labour becomes a privilege for both the employee and the employer.
Eliminating minimum wages means that employment can become a much smoother continuum of activities, rather than the sustenance-or-nothing proposition that it is right now. This opens up a vast range of lower-wage job opportunities, which people could purse when they want to, not actually because they need to. Want to earn £2/hr working in a community theatre? No problem. Want to earn £50/week helping to organise a supper club or being an assistant at a hackerspace? Go for it. Under a minimum wage regime, it's not even legal to employ people in this capacity; under a basic income regime, the distinction between "hobby" and "job" will blur into irrelevance. This means that you will be able to create dramatically more work -- and higher-quality work, because it would only be things that people pursue out of choice rather than necessity -- with a basic income than you could ever have without one.
EDIT -- As to how to handle the transition period: phase in the Basic Income at the same rate that you phase out Minimum Wage and all other forms of income support. A good Basic Income should be roughly equivalent to a good minimum wage; in most countries this works out to be about 50% of the mean individual income. If the Basic Income payments are phased in at the same rate that the minimum wage and other benefits are phased out, then there will be virtually no "losers". High-earners will get a little bump from the Basic Income; low-earners will see their take-home pay unchanged, but they will be increasingly less reliant on their jobs for it. This will put them in a better bargaining position vis-a-vis undesirable jobs, while new types of low-wage desirable jobs will become possible as the minimum wage is reduced. Do this transition over a course of a decade and it should be relatively smooth.
Arguing by your logical suppositions, i claim that what would happen is that jobs that is required for society to function, but nobody wants to do, but is not automate-able (does toilet cleaning count? it might be automate-able...) would mean either an increase in the cost of performing that job, or it doesn't get performed at all at any cost because the benefit doesnt outweight the cost of getting it done.
THis could mean one of two things - if the job is really really essential, then the BI income will eventually not be able to cover the cost of that job (because people would be using their share of BI to cover it). If the job isn't really that essential, then it will end up being undone, much to the detriment of society perhaps.
My basic argument for BI's probable failure (i'd very much like to be proven/convinced otherwise tho), is that the resources invested in the infrastructure to replace labour must make some sort of return on investment (especially if it is made by private individuals). In this case, taxing these automation investments will either make it not worthwhile, or that the owners of these investments will just raise the price of the goods created to counteract the tax, thus raising inflation rate, and doesn't really do any good. The only way it will work is if the investment into automation doesn't "earn" much profit, and produces goods almost at cost. I doubt that would happen at all.
In short; will human nature be able to gracefully handle the transitioning period?