Formally speaking, it's simply a choice you make when you make the first attack on your turn, regardless of what sort of attack it is. It could even be a ranged spell attack; you would still make the choice to be reckless or not at that point.
Starting at 2nd level, you can throw aside all concern for defense to attack with fierce desperation. When you make your first attack on your turn, you can decide to attack recklessly. Doing so gives you advantage on melee weapon attack rolls using Strength during this turn, but attack rolls against you have advantage until your next turn.
This means that if you're, say, an Eldritch Knight/Wild Magic barbarian multiclass, you may wish to Scorching Ray three far off targets, then rage as your bonus action, then action surge so you can haul off and attack twice at some other big boss... You're going to want advantage on the boss, but the time you declare Reckless is when you cast scorching ray, even though the attack rolls for scorching ray itself are not affected.
The only issue with it being a 'stance' toggle is that it may mislead players into thinking they can toggle it off again after their first attack, and 'not be reckless' for the rest of their turn, or avoid the advantage that it gives out, and so on. Using the prompt system would help cement the reminder that it's an all or nothign, and you get it for all your (Str, Melee) attacks that turn, not just the first.
