Styjun

My wife is new to DMing and D&D as a whole.

We had our first game yesterday and one of my friends raised a concern about how she was awarding XP as she was giving the XP to the person that got the killing blow. He felt, and I agree, that this was unfair to the people playing a support role i.e. our Bard and Druid.

Any suggestions on how I should bring this up without it sounding like a criticism or that I’m trying to take over?

By the Book

I'd start by suggesting she review Chapter 8: Running the Game, Experience Points. Hopefully, the part she'll realize what she's doing wrong after reading this:

Each monster has an XP value based on its challenge rating. When adventurers defeat one or more monsters — typically by killing, routing, or capturing them — they divide the total XP value of the monsters evenly among themselves. 

More importantly that just dividing up XP evenly, killing a creature is not the only way to gain XP. It should be possible to get XP with no killing at all. That clearly makes it unfair to award it for a kill - the non-lethal contributions can be just as important, if not more important, than raw damage. Unless the enemies are being played as unintelligent bags of blood and loot, it's even possible to win an encounter without killing anybody.

A Question of Fairness

If needed, you could highlight unfairness of XP-to-the-Killer by pointing out a scenario like this: What happens with something like a rogue sneak attack critical hit or a paladin crit smite leaving an enemy with single-digit hit points, and somebody else taking it out? The rogue or paladin did the vast majority of the damage. Why should the last hit point be more important than the thirty, fifty, seventy, or more that came before it?

Just politely point to the rulebook:

DMG pg 260 (emphasis added):

When adventures defeat one or more monsters ... they divide the total XP value of the monsters evenly among themselves.

Ideally this would have been established beforehand

I've played with new GMs before and these situations can be tricky, and even more so with certain personality or relationship dynamics at the table. It's helpful to establish beforehand a method to address concerns so all players, including the GM, feel comfortable. At a new GM table (and at any table) I ask on the first day of the group meeting how the GM would like the players to address rule concerns. Let the GM guide you on what works best for them.

It's not too late

Just because you've started playing without establishing how to address concerns it doesn't mean you can't still do it. Until you do, let this specific situation lie. It's more important that you establish clear lines of communication that everyone is comfortable with than rushing into solving this specific problem. It's much easier to fix some misappropriated XP than bad feelings at the table.

At your next meeting when you're all at the table, but before game play, ask the GM how she would like the players to address any concerns they have. Listen to what she says and then take her lead. (She may also need time to do research or consider what would actually work best, in this case give the GM the time she needs to figure it out.)

It's worth noting that a new GM may not know what would work best for them, or the group, here. So be flexible if they decide to change it up later down the road.

Be Patient

Learning to GM takes time. There are a lot of rules, and there's a lot to manage. Even experienced GMs slip and need to be corrected. Like any new player, a new GM needs an enjoyable first experience to build confidence and appreciation for the game. Any time a concern comes up ask yourself, "is this so important that it affects my level of enjoyment at the table?"

If it doesn't affect the level of enjoyment and it's just rules for the sake of rules, let it lie. I have found that when I let go of my sometimes rigid expectations around rules the game opens up in new ways and can be more fun than I ever imagined. If it does affect the level of enjoyment address the concern using the method established by the GM.

I am reading into your question a bit when I say that it seems like your personal relationship with the DM causes some uncertainty, or maybe even fear, that she may take a correction the wrong way. I will presume that you know your wife much better than I do, and that your fear in this case is (correctly) based on prior experience.

On one hand, I want to say "Don't be afraid to correct your spouse in a caring and respectful manner." On the other hand, I am fully aware of how silly little things like this can become huge issues when there are gaps in expectations.

However you handle this specific situation, I would also recommend clarifying your spouse's expectations when it comes to DMing. Was she wanting to do things her own way, even if others don't like it and it goes against the formal rules? Even if it's going to result in a major breaking of gameplay? What is the best way to bring rules to her attention? What does she want to get out of DMing?

When you clarify what her expectations are; the question of how to proceed may answer itself; whether that means bringing it up in the moment, or waiting, or whether she fully expects to fly by the seat of her pants and create rules based on whims (and crashing/burning or coming close to it); in which case it might even be fun to suggest the concept that reality itself breaking apart or experiencing strange ripples.