From first week of this invasion, the 2.3 million Palestinians in Gaza stripe (which is basically a prison) have been living with no water, food, electricity, internet, etc. since it was either cut off or the infrastructure destroyed as IDF directly targets those. This is why they've been drinking sea water, gathering rain and sometimes smuggle food through a hole inside their prison wall like this: https://www.aparat.com/v/qOIJ9
So Yemen is currently trying to cut off at least part of the supply reaching the IDF terrorists to even the odds.
Another objective is to increase the firepower needed to shoot down projectiles coming from Yemen headed towards terrorist positions.
You see at any time US Navy + US bases in between + Saudi/Jordanian/Egyptian defenses can only detect a fraction of those projectiles and then they can only engage and shoot down a fraction of what they detect. Increasing the number of warships in the Red Sea can increase the chances of shooting down a higher percentage of those projectiles.
In any case even with the entire NATO navy in the territorial waters they won't be able to protect those ships, nor would any shipping company, insurance company, etc. would be willing to work with Israel anymore and risk being targeted or seized by Armed Forces of Yemen.
Today Yemen Army is no longer a "militia" or "rebels" as US propaganda machines like to call them. It is a solid military known as Ansarollah with over a million highly trained servicemembers that Houthis are only a part of. They possess most advanced weapons specifically drones and missiles and they've shown solid capabilities in air defense (https://www.youtube.com/watch?v=OR5rtB7MuwU). At this point they are categorized as a stronger military compared to half of NATO members[1].
In other words this is not something US is capable of "attacking and getting rid of". Yemen is currently more than capable of sinking the entire US navy, and every easily too. All it takes is an overwhelming number of specialized anti-ship loitering munitions (aka suicide drones) to exhaust the vessels' defenses and do some damage; followed by a handful of heavy anti-ship cruise missiles that would tear a destroyer apart like knife cutting through butter.
After all a destroyer or an aircraft carrier is nowhere near as defended as the US bases in Saudi Arabia or the Saudi Aramco which Yemenis easily attacked and destroyed using the strategy I just explained.
[1] Example: https://www.aparat.com/v/XLndE
@3:44 + @3:48 none of the NATO members except US and France own such long range high speed cruise missiles!
And my favorite @4:04 this is a 10.5 meter medium range ballistic missile weighing 4.5 ton with high maneuverability and hypersonic speed and most importantly pinpoint accuracy. This is something none of NATO members including US can even dream of ever building
Honorable mention @3:09 the lawnmower-136