Enable TLS 1.2 for SQL Server 2016 database mail
TLS1.2 is the only version of TLS considered secure now (March 2019).  It took considerable time and effort to discover that there are 2 essential, additional settings which are required to get this working which are not well known nor well documented, by Microsoft or on the web generally.  The following could save you a great deal of time and effort.
These are the 2 new Registry settings that fixed the problem for us:
Windows Registry Editor Version 5.00
[HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\.NETFramework\v4.0.30319]
&quot;SchUseStrongCrypto&quot;=dword:00000001
 
[HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Microsoft\.NETFramework\v4.0.30319]
&quot;SchUseStrongCrypto&quot;=dword:00000001

This is a reference to the thread where we eventually found this information, buried halfway down the thread: TLS 1.2 in .NET Framework 4.0

Below is the content for a simple executable registry file that I put together that will make the 2 new settings and the settings already shown on the thread above (i.e. this makes all of the necessary Registry settings*):
Windows Registry Editor Version 5.00
[HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\.NETFramework\v4.0.30319]
&quot;SchUseStrongCrypto&quot;=dword:00000001
 
[HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Microsoft\.NETFramework\v4.0.30319]
&quot;SchUseStrongCrypto&quot;=dword:00000001
 
[HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Control\SecurityProviders\SCHANNEL\Protocols\TLS 1.2]
 
[HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Control\SecurityProviders\SCHANNEL\Protocols\TLS 1.2\Client]
&quot;DisabledByDefault&quot;=dword:00000000
&quot;Enabled&quot;=dword:00000001
 
[HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Control\SecurityProviders\SCHANNEL\Protocols\TLS 1.2\Server]
&quot;DisabledByDefault&quot;=dword:00000000
&quot;Enabled&quot;=dword:00000001

Note 1: SQL needs to be restarted for these settings to take affect but it is better to restart Windows since the new settings will affect .NET 4.x generally.
Note 2: In SQL, the SSL-checkbox must be ticked in the mail profile to use TLS1.2.
*Note 3: FYI We ran the free tool, Crypto V2, with the &quot;Best Practices&quot; option enabled before starting on getting this working.  We verified our changes afterwards using the new Crypto version 3.
Hopefully this will save considerable time, effort and frustration for others ;)