One way to check whether SQL Server Local DB is installed is through the record HKLM\SOFTWARE\Microsoft\Microsoft SQL Server Local DB\Installed Versions already quoted in @daniloloko’s reply, however, it is necessary to specify for the RegistryKey if the system is x86 or x64. 
We can do this through the property Is64BitOperatingSystem class Environment to determine whether the system is 64bit or 32bit, then specify which display of the record should be used RegistryView.Registry64; or RegistryView.Registry32; in the object RegistryView.
Follow the software that checks whether SQL Server Local BD is installed:
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using Microsoft.Win32;
namespace SQLServerLocalDBInstalledExemplo
{
    class Program
    {
        static void Main(string[] args)
        {
            RegistryView registryViewArchitecture;
            if (Environment.Is64BitOperatingSystem)
                registryViewArchitecture = RegistryView.Registry64;
            else
                registryViewArchitecture = RegistryView.Registry32;
            using (RegistryKey registryKeyLocalMachine = RegistryKey.OpenBaseKey(RegistryHive.LocalMachine, registryViewArchitecture)) 
            {
                RegistryKey registryKey = registryKeyLocalMachine.OpenSubKey(@"HKLM\SOFTWARE\Microsoft\Microsoft SQL Server Local DB\Installed Versions", false);
                if (registryKey != null)
                    Console.WriteLine("SQL Server Local DB instalado.");
                else
                    Console.WriteLine("SQL Server Local DB não esta instalado.");
            }
            Console.ReadKey();
        }
    }
}
To manipulate the record you need to add the namespace Microsoft.Win32, see also that it was necessary to use the method OpenBaseKey class RegistryKey, this method allows you to open a new RegistryKey representing the requested key on the local machine with the display mode specified on the object registryViewArchitecture.
Sources:
Check if SQL Server is installed on a machine through C#
SQL2012 Localdb: how to check in c# if it is Currently installed?
							
							
						 
Try to check through this record:
HKLM\SOFTWARE\Microsoft\Microsoft SQL Server Local DB\Installed Versions\11.0– gato
The answer to your question has already been given , but if it is possible to migrate to localdb 2016 you do not need anything installed to run !
– John Diego
That is, there is no need to install SQL Server for anyone who runs and is not installed?
– Chirag Geiantilal