Passing the Hash
If you hash the password on the client, you run the risk of introducing a pass-the-hash (PTH) flaw. This is a real, surprisingly common phenomenon that significantly impacts the security of your application. It has been a key security issue in Windows for years: https://en.m.wikipedia.org/wiki/Pass_the_hash
(EDIT: the Windows example above talks about weaknesses in the NTLM challenge-response protocol. A well-designed challenge-response protocol should not be vulnerable to PTH, although these protocols have largely been superseded by transport layer security. See below.)
Essentially, if you hash the password first, the value that you are using to authenticate against the server is actually a hash of that password, and not the password itself. Therefore, an attacker only technically needs to know the hash, and not the password itself. As a result, if the database were compromised, you're now foobarred, because the attacker has all of the credentials they need.
You could work around this by hashing on the client, and hashing that hash again on the server. This means that the database would contain a list of hashes that still need to be cracked before they are useful.
However, this doesn't particularly help your transport security dilemma. Whatever value you send to the server during authentication (the password or just a hash) is the value that an attacker can use to authenticate with against the server. Therefore, a client-side hash always has the same risk profile as the password itself. You're no safer by transferring it as a hash.
As a result, you're best focusing your attention on a good TLS configuration. Make sure that the client will never send sensitive information over a cleartext protocol, and that it only accepts valid server certificates. Consider certificate pinning if available. Configure support for only the most secure protocols and cipher suites on both the server and client.
Alternate authentication methods (all of which have the potential to be invulnerable to pass-the-hash if implemented correctly):
Such protocols are largely unnecessary in modern applications due to the increased availability and interoperability of TLS.