Get rid of the rodents
The decline of multituberculates probably is linked to the rise of rodents. The role of competition in evolutionary history is thought to have been dramatically overestimated, but the relationship between multituberculates and rodents appears to be one of the few exceptions where this is not the case. Multituberculates and rodents show exactly what would be expected in a competition scenario: an inverse-wedge pattern of diversity with increases/decreases in one mirroring the other and different patterns on different continents, in contrast to what you typically see in most cases of so-called competition (i.e., creodonts and carnivorans, most of the taxa involved in the GABI). Most recent studies support the competition hypothesis as the most likely.
As a result, the best way to ensure the continued survival of the multituberculates is to get rid of the rodents. The best way to do this would be to make it so that multituberculates were not as dramatically effected by the K-T extinction in Asia. It's been suggested that the fact that Asian multituberculate communities were comparatively devastated by the K-T extinction is what created opportunities for rodents to diversify in the first place. Unless rodents originated in India as some hypotheses have suggested. It's also an open question as to how much plesiadapiform primates (which are also diprotodont) would compete with multis in the long run, though they seem to have coexisted with them well in North America.
Alternatively, get the multis to move into a niche that rodents can't occupy very well. Rodents can do a lot of things, but they aren't great at everything. They struggle in omnivorous or carnivorous niches because they have no teeth that can be used to shear flesh and they can't easily get one without disrupting their jaw mechanics. Indeed the last multi, Ectypodus, which coexisted with rodents the longest, appears to have been omnivorous or outright carnivorous. This is exactly what you'd expect from a species moving into a niche the rodents can't occupy to coexist.
Make gingkos more abundant
The multituberculate dentition (aside from the more rodent-like taeniolabidoids) is really good at husking fleshy fruits, such as those of ginkgo trees. Rodents are much better at cracking hard nuts due to their gnawing habits (again, except for maybe the taenolabidoids). This is thought to be one potential reason why rodents ended up becoming dominant over multituberculates, as time went on hard nut trees became dominant over fleshy fruited ones. In particular gingkos were hit extremely hard by the K-T extinction and never really recovered. If ginkgos did better in your world and fleshier fruited trees were more common, the multis might have more of an advantage over other diprotodont mammals.
Get the multis established in Australia
One place that took a really long time for rodents to get established is Australia. Rodents have only gotten to Australia within the last six million years or so, and diprotodont marsupials have diversified into so many niches that rodents are unlikely to wipe them out if there is any competition between them (also Australian muroids and muroids in general have very stereotyped anatomy and may have trouble moving into new niches). Get the multis there and they could survive for a long time without interference. It's also really easy to get the multis as the only mammals in Australia, the number of mammal groups that actually made it there was low and their hold was often very tenuous if the local extinction of placental ungulates is any indication.
Just getting the multis to South America and isolating them there might not work. There is an easy opening for multituberculates to get to South America in that they could disperse south like the other major Paleocene groups (marsupials and condylarths) and become established there, and there are possible non-gondwanathere multis in South America itself. However, eventually you'd get caviomorph rodents coming to South America to start the whole thing over again, not to mention potential competition from things like polydolopimorphians.
Or, if you want to be extra cheeky
The New Zealand Saint Bathans mammal is thought to possibly be a multituberculate or a group that was around on other continents at the same time as the multis. Simply have a lineage of multituberculates become isolated on New Zealand, and then you have a landmass where they can diversify to occupy any niches you want without interference from other mammals. It's probably not what you want in that the multis are only dominant in some isolated landmass (Madagascar could work well for this as well), but they'd still be alive.