Generational transfers are generally an easier sell politically than broad-based anti-poverty programs. SSI was deliberately structured to superficially resemble a private pension program, with a payroll deduction and a benefits formula based on how much you paid into it. FDR specifically wanted the payroll tax as a separate, explicit, broad-based tax so that people would see it as something they'd paid into and earned so they'd oppose future attempts to cut or repeal the program, and he specifically wanted the payroll tax to max out at a certain income level to reduce the level (real and perceived) of redistribution in the program so the upper-middle class wouldn't resent it.
As a generational transfer, particularly with a pay-as-you-go mechanism (where current revenue pays benefits rather than being invested to pay future benefits like a private pension program; the SSI trust fund wasn't established until 1977), it was also initially very cheap relative to the perceived level of benefits: only retirees who'd paid the payroll tax were eligible, and those who'd paid the payroll tax for less than a certain lenght of time (five years?) had their benefits prorated, and it took years for the base of beneficiaries to build up to an expensive level. Meanwhile, people approaching retirement age were looking forward to their promised benefits from day 1 of the program and credited FDR for establishing the program.
FDR did have anti-poverty programs (WPA, TVA, AAA, NIRA, etc), but most of them were structured in such a way as both to give him and his Congressional allies flexibility in directing the benefits for political purposes and making sure the beneficiaries knew who they had to thank for their money (not a specific slight against FDR; this is a widespread Stupid Politician Trick, especially in that era, and both parties have been guilty of similar things when they've had the opportunity), and they were generally structured as infrastructure, public works, and economic rationalization (central planning, rationing, and price controls were at the time widely believed to be more efficient than a lightly-regulated market economy) projects that could be argued were real jobs doing things that benefited taxpayers as well as the direct recipients of federal money, rather than being undisguised and unmitigated handouts (which tend to be unpopular, especially when taxes are higher than people are used to (some the largest tax increases in US history had passed under Hoover's administration, and FDR raised taxes further) and when the transfer programs aren't things that people are used to).
Then there's the effect on unemployment. SSI, in the short term at least, reduces unemployment by encouraging people to retire earlier and leave the job market. A Basic Income program would also give people marginally in the job market an incentive to stop looking for work, but it would do so in a way that looks bad (discouraged workers resigned to living off the dole) rather than good (people able to afford a comfortable retirement after a lifetime of hard work).
If FDR decided to pursue a Basic Income program instead of SSI, or indeed as a replacement for the bulk of the New Deal, he probably could have gotten it passed into law. However, it would have been much less popular than SSI, perhaps to the point that a repeal a few years down the road would be plausible, and if it had also replaced the WPA, TVA, and similar programs, it would have also cost him a lot of his ability to use New Deal money to reward his political allies, thus reducing his political clout significantly.
In short, if you want an Alf Landon or Wendell Wilikie Presidency, this is probably one of the better PODs to use to achieve it.