Every player of every class, whether ranged or melee, magicka or stamina based, is encouraged (especially in veteran ranks) to learn to use the all-powerful block, dodge, sprint, sneak, cc-break and interrupt.
In fact, these skills are known as 'finesse' actions in game, and their proper application is a major focus of difference in player skills.
But as many players have noted, a player who is using melee weapons exclusively or at least primarily, suffers more due to having to share this resource. Magicka players can reserve their stamina bar for finesse skills, and focus their class and stave abilities on the magicka bar.
Endurance
As others have suggested, one solution to this imbalance is an 'Endurance' bar. If you've ever been on a horse and noticed that small, thin little stamina bar below, keep that image in your mind.
Endurance would be a derived attribute, no longer completely interchangeable with stamina. Now it would be a combination of magicka, health, and stamina, so that all builds have access to this important attribute. Health and stamina having more of a benefit to this attribute pool than magicka.
Any passives or abilities which currently effect stealth costs, sprinting costs, blocking costs, dodging costs, interrupt costs, or cc-break, would instead effect the cost from this new endurance bar.
Potions could be built to focus on adding to this bar, as well as providing passive regen in heavy armor, enchants and set bonuses.
How this affects stamina?
This would completely remove the cost association with finesse abilities from the stamina pool. This would mean that stamina would be set aside completely for the use of weapon and armor based abilities, as well as any guild lines, similar to it's magicka counterpart.
And it would allow magicka and health focused builds to benefit from their point allotment, without having to invest in stamina to use abilities otherwise unrelated to their build.
Too long, didn't read
Make a new bar, call it something like endurance, make block and dodge drain it instead of stamina, and solve a major crux in your problems with stamina builds competing against magicka builds.
Ruze Aulus. Mayor of Dhalmora. Archer, hunter, assassin. Nightblade.
Gral. Mountain Terror. Barbarian, marauder, murderer. Nightblade.
Na'Djin. Knight-Blade. Knight, vanguard, defender. Nightblade.
XBOX NA
Ruze is a veteran of the PC Beta, lived through the year one drought, survived the buy-to-play conversion, and has stepped foot in the hells known as Craglorn. He mained a nightlbade when nightblades weren't good, and has never worn a robe. He converted from PC during the console betas, and hasn't regretted it a moment since.
He'd rank ESO:TU (in it's current state) a 4.8 out of 5, loving the game almost entirely.
This is an multiplayer game. I should be able to log in, join a dungeon, join a battleground, queue for a dolmen or world boss or delve, teleport in, play for 20 minutes, and not worry about getting kicked, failing to join, having perfect voice coms, or being unable to complete content because someone's lagging behind. Group Finder and matchmaking is broken. Take a note from Destiny and build a system that allows from drop-in/drop-out functionality and quick play.