How can the Client hold down a key that plays the animation without it constantly playing?

AnimationTracks have an IsPlaying property. Create your animation track variable (playAnim) outside of the event and you should be able to use this property.

local UIS = game:GetService("UserInputService")
local player = game.Players.LocalPlayer
local humanoid = player.Character.Humanoid
local anim = Instance.new("Animation")
anim.AnimationId = "http://www.roblox.com/asset/?id=3398376417"
local holdingSKey = false

local playAnim = humanoid:LoadAnimation(anim)

UIS.InputBegan:Connect(function(inputObject) 
    if(inputObject.KeyCode==Enum.KeyCode.S)then
        holdingSKey = true

        while holdingSKey do
        --Check if the animation is playing, if it's not, play the animation again
        if(not playAnim.IsPlaying)then
                playAnim:Play()
                wait(0.2)
        end
        end
    end
end)

UIS.InputEnded:Connect(function(inputObject)
    if(inputObject.KeyCode==Enum.KeyCode.S)then
        holdingSKey = false
    end
end)

You could simply use the AnimationTrack.Looped property for this case:

local UIS = game:GetService("UserInputService")
local player = game.Players.LocalPlayer
local humanoid = player.Character.Humanoid
local anim = Instance.new("Animation")
anim.AnimationId = "http://www.roblox.com/asset/?id=3398376417"
anim = humanoid:LoadAnimation(anim)
anim.Looped = true

UIS.InputBegan:Connect(function(inputObject) 
    if(inputObject.KeyCode==Enum.KeyCode.S)then
        anim:Play() -- This will keep looping until stopped
    end
end)

UIS.InputEnded:Connect(function(inputObject)
    if(inputObject.KeyCode==Enum.KeyCode.S)then
        anim:Stop() -- Simply stop the looping
    end
end)