It is not clear what you are doing that makes the final dimension too large. But guess is that you are not using Save For Web. Do not just Save as PNG. Use the Save for Web (may be under File > Export > Save for Web in newer AI) when needing a JPG, PNG, or GIF.
Under the Save for Web tool, you can control the precise final size in pixels even if the AI Artboard is not accurate.
BONUS – Photoshop has the same feature.
##---update I am not giving up on you, @Lauren ;)
I took your banner and Save for Web (as 300x60). As you can see, it is just 26.7K (not too large). And at 100% size – it is as sharp as can be using that font.
Would you consider this "blurry"? Your Save for Web still produces a large file? Have you ever been able to save a 300x60 pixel banner that you would consider sharp? If so, can you share it with us?
