First of all, you can use custom fonts on websites. There's no need to upload a textual logo as image unless you're using special effects like gradients on the letters.
Now, as for your problem: just export the file from Illustrator as SVG. It's a vector file format supported by every browser.
You don't even have to worry about the size, just embed it like this:

The browser will resize it while keeping the aspect ratio, and since it's a vector image, you can set arbitrary sizes without blurring or lost details.
As for your current workflow, resizing a logo in Photoshop should not visibly damage a big file, as long as the target size is exactly the one you're going to use on your website.
If you resize the logo to a width of 100 pixels, then display it 300pixels wide on the website, of course it'll look like crap.