Are you using http:// or are you using the secure https:// log-in?
You can use this link to see if your Flash is up to date. http://www.adobe.com/software/flash/about/ If it’s not up to date then this is where to download and install the latest version of Flash. http://www.adobe.com/products/flashplayer/
Please try using the browser uploader. Click the insert image button and then down below the "select" button is a string of text and in that is a link to the "browser uploader." Click that and see if the non-flash browser uploader works for you.
These are the troubleshooting steps we take whenever any issues like this arise:
clearing your browser cache and cookies > http://www.google.com/support/accounts/bin/answer.py?hl=en&answer=32050
using the browser with all extensions or add-ons disabled
updating the browser to the latest stable version (not a buggy BETA browser version) http://browsehappy.com/
confirming third party cookies is enabled http://support.google.com/accounts/bin/answer.py?hl=en&answer=61416
resetting your router and modem and rebooting your computer
then logging in using the secure https:// log-in see http://en.support.wordpress.com/https/
trying another browser
If those do not work we need to know how are you connecting to the internet and to WordPress.com? The way you connect to the internet (mobile, satellite, DSL, dial-up) and to your blog, and how many proxy server jumps etc. it takes to connect can cause problems. There maybe be problems with your ISP and/or with the proxy servers.
And exactly which browser and version of it are you using. If you don’t know click here to find out > http://whatsmyuseragent.com/