What wavelength should I use for my ray gun?

You don't need a ray gun. Just shoot him with a normal gun, and when your Menace becomes intangible, watch him fall through the ground to the center of the Earth.

Once there, he can either become tangible and die from heat and pressure, or remain intangible for as long as his air supply lasts before suffocating.

Problem solved!

Simply scan through the frequencies. When the baddy flinches, turn up to 11 and fry him.

From a practical engineering POV, near infrared lasers have the greatest range in atmosphere, which allows you some latitude in targeting the bad guy. This is important since lasers are notorious power hogs, and the conversion efficiency from generator or chemical reaction to light is very poor. You will need a large installation, waste heat rejection, the optical train and a high quality mirror. A US Army experimental anti aircraft laser installation requires a tactical transport truck, since the power generator is a 300Kw turbine engine from a scout helicopter.

Dynetics and Lockheed design for a laser weapon

Longer wavelengths are possible, including microwaves, but the size of the mirror becomes correspondingly larger. The microwave Active Denial System fires a beam which causes pain, but only non lethal amounts of energy is delivered to the target. Obviously a more powerful microwave generator could be installed, but the size of the vehicle could be adversely affected

Vehicle mounted ADS

For the best results, you may need to go for a laser weapon mounted on an aircraft. This provides elevated sightlines for sensors and the laser beam, the ability to rapidly reposition the weapon, and the ability to tap into a large and powerful turbine engine for energy

F-35 laser mount for the F-35A and C models. Marines are out of luck

Super science is pretty expensive these days.....

Close to visible light seems best

Visible light is chosen by the eyes for a reason. They are fast, abundant and have a lot of energy. That means you don't need incredibly sensitive eyes. Maybe ultraviolet is better, someone might know this better, but visible light packs a punch. If you go higher in wavelength you get into trouble with passing through the body, like x-rays. So lots of energy is lost.

But your guy is transparent, which means visible light passes through, not imparting it's energy. So you need to have something around that spectrum that imparts it's energy and doesn't pass through him. But is he transparent in those spectrums? We don't know.

But there is the problem. A beam of light from a handheld flashlight doesn't do much, unless you can focus it on a tiny point for a little while. Maybe he's translucent because he's allergic, so a focused beam would do more damage, but unlikely. So expecting more than focussing the beam and burning him would be difficult.

Maybe he's overly sensitive to radiation? Humans are pretty good with it as creatures.

But a laser isn't the best choice, is it? Any harmful electromagnetic wave will do

A blue laser should do. The villain appears to be able to become transparent to most wavelengths except visible light; and the only reasons visible light is innocuous is that its frequency is too low and its energy density is too small.

But concentrate enough a small amount of energy and you get a lot of power. A 7 watt blue light laser can ignite paper (and render instantly, permanently blind).

Concentrate enough such lasers using an appropriate crystal or mirror matrix, and you can literally burn the villain to death.

(Frame challenges removed due to clarifications in the question)

According to some calculations done by PH.D. Rebecca Thompson of the American Physical Society, a headshot from a grid of regular laser pointers as weak as 1 kw in total would probably kill a person faster than they could get out of the way. The HELLADS program is a DARPA project that developed a 75 kW laser that weighs only about 375kg. This means that a 1-2kw weapon based on the same technology would be hand portable.

The exact wavelength of this laser is not public knowledge, but we do know that it is a solid-state laser. The older generations of military grade solid-state lasers they were based off of emitted at a wavelength of about 1 micrometer making them near infrared beams; so, if I had to guess, I'd assume it is likely in the same approximate wavelength.

The HEL beam is about 10 times lighter than previous generations of lasers for its stopping power; so, it's safe to say your heros will care more about replicating the engineering behind it than what exact wavelength it is operating at.