This is why I liked Mary Robinette Kowal's approach, of encouraging anyone who could to obtain membership and then vote however you would have anyway. She even bought some memberships and gave them to those who couldn't afford them, regardless of how they intended on voting. The idea being that the more people who were voting out of a legitimate conviction, the harder it would be for any group to artificially influence or rig the vote.
On the wider participation point it's worth noting that the three years of Sad Puppies campaigns have led to a massive increase in the number of Worldcon members, with record numbers the past two years. Whether that's due to people signing up to vote for one of the Puppy slates, to vote on the anti-Puppy slate or simply because the additional publicity this year meant that people finally realized that anyone can pay the gatekeeper fee and nominate/vote for the Hugos, the result has been dramatic. This year it took 2649 votes for Three Body Problem to win in the final runoff for best novel... in 2010 (so hardly a long time ago) the top two drew with 380 each. That year Patrick Nielsen Hayden won the best long form editor Hugo with 200 votes on the final run-off. This year Toni Weisskopf came second with 1216 (over six times as many) but was crushed by the slate to no-award her with 2496 (over twelve times as many). If the best way to prevent groups from "gaming" the Hugo Awards is to increase participation then the various Puppy slates should be applauded for being the driving force behind getting so many new members.
Looking at Kowal's point specifically there's a pretty obvious issue with it, albeit not one that I think she foresaw. Let's say Beale or Larry Correia made the same offer on their site; they'll buy X number of memberships for anyone who replies/messages them asking them, no string attached, vote for whatever you want. I suspect if you were to able to isolate out those votes you'd find that they resembled the Rabid/Sad Puppies slates fairly closely. People tend to attract like-minded people and you can presume that someone who follows an author who ever talks about what they think of fandom is going to have a similar view.
(One caveat to the above; if you look at the statistics then the Sad Puppies seemingly didn't vote as a slate. While the influence of the Rabid Puppies is obvious and one can see around 600 people or so who followed that slate entirely the Sad Puppies vote appears to be much more fragmented. Looked at statistically there only really seem to be two slates that actually worked as slates; the Rabid Puppies and the "vote for everything else, then no-award then the puppies" slates)
There's an ongoing rule update that may come into play in a year or two which should lower the impact of Slates, at least for nominations. Rather than the system that is currently used where each voter can nominate five works in each category and each nomination is counted equally the system can change so that each person has "one" vote but can split it; to look at it percentage wise if you nominate a single work it gets 100% of a vote. If you nominated two they each get 50% of a vote etc etc. So if a slate wanted to nominate five works to completely dominate a category then each vote for the whole slate would be worth only 1/5th of someone who only wanted to nominate one work.