I'm not aware of any way to downgrade a client. Did the client connect to a 7.1 server and upgrade automatically? I thought that would require installing the new 7.1 client but I haven't tried that yet.
Regardless, you can download the 2017 May (7.1) client by navigating to https://www.dynatrace.com/support/downloads/ and going into AppMon.Then just install the required client (might want to uninstall the 7.1 one first if you won't need it).
Because you have 2 different environment you would need 2 different clients. I would recommend in this case for the 7.0 server to use the portable client as it can run in the folder it is extracted into.
As has been alluded to, the 2017 May 7.0 client will not work to connect to the 2018 April 7.1 servers. Although you may still get the initial connection to the server to perform the upgrade to the client, they really are not all that compatible. (Actually the upgrade message informs you that by proceeding to update, you'll no longer be able to connect to 7.0 servers).
The upgrade between 7.0 and 7.1 is different to say an update from 7.0 to 7.0.9. Updates are much simpler to remove from a client. Upgrades on the other hand will require a re-installation.
You basically have three options: