Having dodos survive into the modern day is a very difficult proposition, if not completely impossible for a timeline as we know it. Let's break it down into what we understand about the dodo, and why it died out.
Contrary to how it is generally portrayed, what we know about dodos (which is a lot less than one would expect given the fame of the animal: partial remains from four specimens brought to London, some written anecdotes and drawings from sailors, a couple of paintings of animals in menageries, and some subfossil remains from a single swamp in Mauritus) suggests that they weren't nearly as dysfunctional as commonly made out to be. For one, they don't seem to have been (exceedingly) fat and sluggish birds in their natural habitat; their leg bones indicate at least some capacity for fairly fast movement, and the obesity seen in depictions of the few live specimens brought to Europe is nowadays believed to be the result of overfeeding of the birds. While they seem to be gluttonous when faced with fruit as indicated by written sailor records, the fact that Mauritus has an annual dry/wet cycle indicates that this was an adaptational feature to encourage birds to gorge on fruit when it fell at the end of the wet season, giving them fat reserves to rely on during the drought. Birds in a captive situation with reliable food would not have to stop gorging themselves, or go through a starvation period, making them get as fat as those depicted in the more famous paintings of specimens imported to European menageries.
Their lack of fear towards larger predators is a bigger issue, and ties into the big one with dodos: their lack of adaptation to deal with non-native insular animals. Pressure from actual humans on dodos was fairly minimal, in the sense of most island endemics - while they were easily hunted by sailors, their flesh was universally regarded as distasteful, and no records exist of sailors making attempt to collect their eggs or young (as was the case with seabirds on islands where they nested en masse). The main problem, and the main reason they seem to have gone extinct so quickly as they did, was the introduction of various invasive animals to Mauritus by sailors that either killed the adult birds (dogs), ate chicks (cats), or most importantly, ate their eggs (pigs and rats). Not much is known about dodo reproduction, but they seem to have followed the common trend for flightless island birds of only one or two offspring per nesting season. While young dodos seem to have grown very fast compared to other flightless island endemics (growing to independence seemingly before the end of their first wet season, in time for the drought), tiny clutch sizes coupled with a lack of any defenses of nests from predators (as well as cats and dogs hunting young) meant that eventually there was simply no way for the birds to reproduce effectively. Coupled with the rest of the island's ecosystem being massively disrupted at the same time, and the recipe makes for a rapid extinction.
Moving into how this ties into their extinction (or how it could be prevented), there really isn't a way for this to be avoided unless the Age of Exploration as we know it is averted. Any sort of naval exploration would lead to the introduction of rats (and by extension ship cats) to the island, and setting up of stations or bases would invariably lead to some other animals being brought along to allow for some self-sustenance. Unless Mauritus simply isn't discovered until after some form of conservation ethos develops to prevent this (which is pretty much ASB, especially when considering both the value of the naval trade routes in the region and the dodo's role in developing the idea of human-caused extinction among the public), they are going to be killed off on Mauritus at some point following human discovery.
There is one potential out, though - a captive population being established outside of Mauritus, presumably in some form of royal menagerie. There is precedence for this (and not only with the specimens brought back to Europe - a painting exists of a dodo in a menagerie alongside native Indian birds that has been linked to the menagerie of the Mughal Emperor Jahangir at some point in the 1620's to 1630's; this is incidentally generally believed to be the most accurate depiction of a healthy dodo in existence), so in theory it is physically possible for a breeding population to have been transported to a suitable menagerie and maintained for some centuries until conservation ideology develops and some can be reintroduced to Mauritus. This sort of conservation has actually happened once, with the
Pere David's deer; the problem now is with the dodo's (presumably) specialized reproductive systems and diet, which would make captive husbandry difficult in a non-native environment. This could in theory be worked around, but it would be hard to justify within the bounds of realism.
In the end they do have
a shot at survival (if only in captivity in a foreign land), but realistically there is sadly not much of a way to keep them alive unless one rewrites the history of maritime exploration and conservation.