There's a problem with my open/close gui?

First of all, you're making changes to the Frame that's inside the StarterGui, which is wrong, once a player joins the game everything inside the StarterGui will move into their PlayerGui. So you'll need to get the GUI from their PlayerGui and make changes to that for them to see those changes.

You could just do this which is a much simpler method.

local Player = game.Players.LocalPlayer
-- You'll most likely need to change these variables to where your GUIS are located inside the Player.
local ScreenGui = Player.PlayerGui.ScreenGui
local Menu = ScreenGui.menu
local Frame = ScreenGui.Frame

Menu.MouseButton1Down:Connect(function()
    Frame.Visible = (not Frame.Visible)
end)

Try this

local function open()
    local menu = game.StarterGui.ScreenGui.menu
    local frame = game.StarterGui.ScreenGui.Frame
    frame.Visible = true        
end

local function close()
    local menu = game.StarterGui.ScreenGui.menu
    local frame = game.StarterGui.ScreenGui.Frame
    frame.Visible = false
end

local menu = game.StarterGui.ScreenGui.menu
local frame = game.StarterGui.ScreenGui.Frame

menu.MouseButton1Down:Connect(function()
    if frame.Visible == true then
        close()
    else
        open()
    end
end)

I haven't tested it out, so hopefully it works!

Don't forget to upvote and make this solution IF this works :)

The two notorious mistakes I can see is that you didn't use playergui than startergui because if you don't the code will function when the player has respawned causing errors and confusing the script. Also, use a not because it's quicker and a time saver:

local function open()
local player = game.Players.LocalPlayer
    local menu = player:WaitForChild( "PlayerGui" ).ScreenGui.menu
    local frame = player:WaitForChild( "PlayerGui").ScreenGui.Frame
    frame.Visible = true        
end

local function close()
    local menu = player:WaitForChild( "PlayerGui").ScreenGui.menu
    local frame = player:WaitForChild(" PlayerGui").ScreenGui.Frame
     frame.Visible =not frame.Visible
    end
end

local menu = player:WaitForChild( "PlayerGui").ScreenGui.menu
menu.MouseButton1Down:Connect(open())
menu.MouseButton1Down:Connect(close())