The confusion arises because some people assume that Basic Income will give tenants (and prospective tenants) more money than they have now. But this isn’t necessarily the case at all, because Basic Income is not more money on top of what people have now. It is money people will get instead of existing welfare payments and instead of existing tax allowances.
Basic Income is about giving people security of income, not more income.
The government could decide to implement a Basic Income system that will be more redistributive than existing tax and welfare arrangements (It doesn’t have to be!), but even in that case, a simple bit of demand and supply analysis suggests that only a proportion of poorer people’s extra incomes is likely to be taken up by rent. So they could be better off and have higher discretionary incomes, thanks to the extra taxes paid by richer people — such as their landlords.
(Oh — and by the way, that’s not really what ‘rent-seeking’ refers to!)