I just recently updated to Unity 5.x and noticed that the Asset Store doesn't work for me anymore. I have Unity 5.4.1f1 on macOs. Either going to the Asset Store from Safari and clicking "Open in Unity", or directly from within Unity (⌘ + 9), I always get an empty Asset Store tab.
Opening Dev Tools or moving the mouse arounds shows me that the page is actually loaded but just doesn't render. While I haven't yet found how to fix this, there's a workaround.
You can also browse the Asset Store by entering keywords in the Project tab search bar and selecting Asset Store. Select an Asset that seems to be in the package you're looking for and the inspector tab will show the details and allow you to import the whole package directly without opening the Asset Store.
|Alternative Asset Store search|
Now, if like me you haven't downloaded a package for a while you probably haven't accepted the latest Asset Store "Terms and Conditions" disclaimer. Unfortunately this alternative access to the Store doesn't give you the opportunity to accept those new Terms and you will get the following message "The server was unable to build the package. Please re-import". What you will need to do is install an older 4.x version of Unity which uses a different rendering method for the Asset Store and was working fine for me. With this older version, download any package from the Asset Store, accept the disclaimer and re-install the latest 5.x version. From now on, you will be able to use the alternative way described above to download packages.
Hope this helps. And if you know how to fix the rendering issue on macOS and Unity 5, please let me know.