The two previous answers demonstrate how to use Canvas and ImageData. I would like to propose an answer with runnable example and using an image processing framework, so you don't need to handle the pixel data manually.
MarvinJ provides the method image.getAlphaComponent(x,y) which simply returns the transparency value for the pixel in x,y coordinate. If this value is 0, pixel is totally transparent, values between 1 and 254 are transparency levels, finally 255 is opaque.
For demonstrating I've used the image below (300x300) with transparent background and two pixels at coordinates (0,0) and (150,150).
Console output:
(0,0): TRANSPARENT
(150,150): NOT_TRANSPARENT
image = new MarvinImage();_x000D_
image.load("https://i.imgur.com/eLZVbQG.png", imageLoaded);_x000D_
_x000D_
function imageLoaded(){_x000D_
console.log("(0,0): "+(image.getAlphaComponent(0,0) > 0 ? "NOT_TRANSPARENT" : "TRANSPARENT"));_x000D_
console.log("(150,150): "+(image.getAlphaComponent(150,150) > 0 ? "NOT_TRANSPARENT" : "TRANSPARENT"));_x000D_
}
_x000D_
<script src="https://www.marvinj.org/releases/marvinj-0.7.js"></script>
_x000D_