If you have decided to go on with your Frankenwallet having restricted Internet access, rather than no Internet access at all, you will able able to use the applications in this section which depend somehow on Internet connectivity. Therefore we're calling this the "cool" environment (since it's not quite cold).
➤ Internet browsing should be prohibited on "cool" environments, just as you would not expect to browse on a "cold" environment.
However (and in case you have skipped to this section), you should still follow the instructions on this page (securing the browser) to curtail any automatic or institutionalised ("spyware") activity from Firefox (the default and only browser which comes with Ubuntu).
It is against these recommendations to mix the security levels of both the "cool" and the "cold" environments… for instance:
➤ Do not keep unencrypted private key files on any machine that also runs any Internet dependent software like a UI node wallet (e.g. Daedalus on Cardano). The chances of losing the assets secured by those key(s) are too great.
➤ Conversely, do not permit routine Internet access from any machine which stores those keys, or on which you enter the high security password for your most sensitive files (e.g. any that stores your wallet passphrases), at the risk of compromising either those keys or that password.
If you think you will need both a "cool" and a "cold" environment, you should create two Frankenwallets: just please be sure to follow the cardinal rule of never entering your "cold" Frankenwallet password into any Internet connected machine... which includes the "cool" Frankenwallet.
➤ i.e. The two Frankenwallets at different security levels should have separate "high security" passwords.