This is how I would have done the coloring (on the right):
You may be right that there might be too much red, but if you look at the overhead lighting, the original image at least has a proper white point. When I brought the untouched TrekCore image
into Photoshop, you'll notice that it's pure 100% white: the RGB values are 255 across the board and there's no hue or saturation.
However, in your altered right side, your RGB values are off. What should be pure white is tinted blue. The red is down to 245, your hue is at 180 and your saturation is at 4%. The only way to know what the correct color is supposed to be for that set and those costumes (and that's not necessarily what the filmmakers intended mind you!) is to see a 18% gray card under those same lighting conditions and set your gray point off of that.
To maintain the whites, you have to hit Ctrl+L in Photoshop and choose the gray eyedropper tool and look for what should be a neutral gray in the image, like the silver wall segment above the redhead on the left. Zoom into the image about 400%, you'll see different colored dye clouds or clumps on the film negative. Try to avoid blue or greenish/yellow areas as those will skew the color too far in either direction.
That should accomplish what you're trying to do better. But definitely start with the TrekCore jpg. The image you're using is cropped, terribly softened and covered in compression artifacting.