You can still sprint when stamina < 0. What have I missed?

You check once, that if statement will never run again. Ever. Running will be true 100% of the time.

Running is set to false in the else statement you will never reach.

This works, i have tested it. It will also only sprint if you HOLD shift.

local mouse = game.Players.LocalPlayer:GetMouse()
local player = game.Players.LocalPlayer
local up = false

local function onKeyDown( key )
    print("Key:", key, " Code:", string.byte(key))
    if string.byte(key) == 48 then
        up = false
        hum = player.Character.Humanoid
        hum.WalkSpeed = 25
        script.Parent.Stamina.Value = (script.Parent.Stamina.Value ~= 0) and script.Parent.Stamina.Value - 1 or 0 -- This causes it to only take your stamina if you have greater than 0. So it does not go negative.
        script.Parent.Size = script.Parent.Size - UDim2.new(0,2,0,0)
        while (script.Parent.Stamina.Value > 0) and not (up) do -- Only loops if Stamina is greater than 0 AND you are holding shift.
            script.Parent.Stamina.Value = script.Parent.Stamina.Value - 5
            script.Parent.Size = script.Parent.Size - UDim2.new(0,2,0,0)
            wait(0.1)
        end
        hum.WalkSpeed = 16  -- When the loop ends(release key/run out of stamina) it will reset it
    end
end

mouse.KeyDown:connect(onKeyDown)
mouse.KeyUp:connect(function(key) -- Anonymous function, you need to know when the key is released so it only works when you HOLD it.
    if string.byte(key) == 48 then
        up = true
    end
end)