If the minimum wage had kept up with inflation since 1968 (the high point of the minimum wage in constant dollars) it would be a little over $17hr. now, based on CPI data from the Federal Reserve. The Consumer Price Index (inflation, as commonly stated) is probably a bit high as an indicator, since it does not take into account consumers shifting around what they buy to the extent they can. The (Personal Consumption Expenditures index) is probably somewhat more accurate, and a bit lower. Still, however inflation is measured, the minimum wage has not kept up, even approximately.
If a UBI is phased in, I think the minimum wage should be set to whatever it would have been without the UBI, minus the hourly amount of the UBI, and at some (decades) later point, eliminated entirely. Say if the minimum wage would have been $12/hr. and the UBI is $1000/month, then the minimum wage with UBI would be set to somewhere around $6.00/hr. This would be perceived by employers as a reduction in the cost of hiring an employee, benefitting commerce.
Employers would notice one big difference however -- they no longer are their employee's only source of income, and they'll have to treat them better, however that gets manifested. This is in many ways a good thing, because while employers/owners do accept the majority of the "long term" financial risk, they almost always have far less immediate risk when negotiating with an employee. This kind of asymmetry in an economic relationship takes away a lot of the "freely entered transaction" aspects of it, and that's economically not a good thing.
So how does it get paid for? Well, for those who work they wouldn't pay a thing at very low income levels. At higher, but still quite low, income levels the IRS would start taking a bit of that UBI back, and that percentage amount would increase the more you made. (At no point, however, would working more get you less money, like welfare can do now.) At a certain income level, the IRS is taking it all back so it's financially neutral relative to the current tax setup. Above that level, you would be paying it all back and then some. If you work in software, for example, there's a good chance you do that now, especially if you have moved along in your career.
But you make UBI and all other non-healthcare assistance an opt-in either or proposition. If you are on some form of public assistance (again, not healthcare) you are given a choice, if you are provably a citizen -- take the UBI and we'll stop hassling you, or can choose to continue the way you are now, but not both. Making it a choice will largely silence the complaints about something being taken away. That will make it possible to incrementally sunset the existing welfare programs -- people will choose to on their own.
If you are a high-income taxpayer, then you are paying the leftover amounts for either the current public assistance scheme or the UBI one, for any given hypothetical beneficiary. One pays more people and is highly efficient (cutting checks and withholding wages, plus validation of citizenship, age of majority, and a pulse) and the other has a lower payout to the recipient, but a shitload of inefficiencies and government intrusion and perverse incentive associated with it.
If it is paid for (at least to the extent the government currently pays for things) and not funded by printing dollars, it will not increase the money supply, so it won't be inflationary in that sense, except as a short term "shock". What it may do is impede commerce/work by reducing the penalty for not working, but this is not a comfortable amount for anyone to live on, much less build a future on. Conversely, the reduction of the minimum wage as perceived by employers might increase commerce and consumption, mostly at the lower income levels. The demand curve for employment will shift, but the supply curve will shift too, likely not as much.