The main thing that promoted trench warfare in WWI were the ability to move and supply a large amount of soldiers to create a solid series of lines of defense across an entire front e.g. railways, canned food, the predominance of certain weapons - machine guns firing enfilade could mow down the majority of an almost unlimited amount of enemies attacking at once, making it easy to defend even when significantly outnumbered, industrialization to mass produce the weapons/ammo needed in sufficient numbers, and no suitably advanced/developed way to reliably bypass such lines of defense that didn't end up stalemated - high profile failures like Gallipoli can also make such attempts to bypass the front lines or open new ones politically unfeasible even if they would be militarily advantageous.
So in terms of dealing with ground based hardware you would want a relatively cheap to manufacture weapon (or combination of weapons) so your defensive lines can have plenty of them, that they are easy to use (so you can arm up lots of conscripted civilians without spending an age training them), and can take out any ground units that advance rapidly, but is heavy enough or requires a short set up time before firing so it can be moved to a new defensive position relatively quickly, but it can't be used when assaulting (unless you are Jesse Ventura/Arnie), with decent fire rates so defenses can't be overwhelmed numerically very easily, and with relatively cheap and easy to produce ammunition.
Note that I don't think the literal trenches of WWI are likely in the future, something more in the elastic defense mold would seem to trump it - lots of small defensive positions that are abandoned if the enemy comes in strength with enough defense in depth to let the enemy extend themselves while giving time for a counter attack to be organized. This sort of dispersed defensive set up tends to make artillery and other forms of massed fire less effective as well.
Assuming that the next problem is to deal with other modes of bypassing these defensive lines - paratroopers and other aerial threats, and naval and amphibious attacks. In the air I think it should be feasible to create a similar stand off - with enough SAM threats on the ground in the layers of defense and enough AAM capability to rapidly move to counter a concentration of enemy air power around a developing attack you could see a stalemate where after long enough the two sides might largely stop producing bombers or close support aircraft (replacing them with more SSM launchers behind the lines to work as guided artillery instead of ASM). You could even see a scene describing the first (and only) attempt at a paratrooper drop early in the stalemate having them all wiped out before even leaving their aircraft (paralleling Gallipoli).
Depending on the location/nations involved it would also need a similar naval standoff to develop, with both sides being able to defend their supply lines and coasts, and some way to make it so the defensive side is favoured in naval battles. For example one side could be following a fleet-in-being doctrine if outnumbered (but not by too much), as long as the air defenses of the port(s) in question are solid enough (a lot of SAM sites and plenty of anti-missile defenses). Alternatively long range amphibious landings might be ruled out as too difficult to keep supplied safely for long enough to make a breakthrough, and smaller ones as being strategically useless as the defender would quickly set up a new line of defense, so the attacker would expend a lot of resources for little practical gain.
Of course such situations tend to generate lots of innovation to try to break the deadlock - taking WWI as an example the obvious one was tanks, also using planes for a myriad of roles - scouting, strafing, bombing and air supremacy, massive amphibious landing, doctrinal changes such as advancing under a creeping barrage of artillery, massive sapper operations, etc. The problem with these tends to be they take time to develop to the point they become truly effective - in this example some of them took about 20 years after learning the mistakes and improving the engineering from WWI and maybe even then a couple of years of "live fire" tests to reach their full potential.
As you suggest something like a railgun with a reasonably large battery in order to power the magnets could be an option for a weapon that is better suited to defending than attacking, especially if it has some target spotting/aim assist built in so even a novice can use it to attack any moving targets then this could act as the base for the ground side of the stalemate, and then some missile targeting advance that makes it easy to clear the skies and seas of enemies, and maybe tanks as well, without making much difference to targeting small ground targets (i.e. people in foxholes/bunkers/buildings) would do most of the rest.
