Manage connection without 'using' statement
I can understand why you want to remove the clutter from your code - but sharing the instance of MySqlConnection isn't a good idea. As soon as multiple threads are trying to use the connection at the same time, things are going to go wrong.

The connections are already pooled for you (See here) so you creating and disposing IDbConnections is cheap and easy.

I think you should create a connection factory:

public interface IConnectionFactory
{
    IDbConnection GetOpenConnection();
}

public class ConnectionFactory : IConnectionFactory
{
    private static readonly string connectionString =
        ConfigurationManager.ConnectionStrings["ConnStr"].ConnectionString;

    public IDbConnection GetOpenConnection()
    {
        // Is there an overload to automatically open the connection?
        var connection = new MySqlConnection(connectionString);
        connection.Open();
        return connection;
    }
}


Then your base repository can do:

public abstract class BaseRepository
{
    protected readonly IConnectionFactory connectionFactory;

    public BaseRepository(IConnectionFactory connectionFactory)
    {
        this.connectionFactory = connectionFactory;
    }
} 


Then all of your actual repositories can do:

public class MyTableRepository : BaseRepository
{
    public MyTableRepository(IConnectionFactory connectionFactory) : base(connectionFactory) {}

    public List&lt;MyTable&gt; GetAllMyTable()
    {
        using (var connection = connectionFactory.GetOpenConnection())
        {
            return connection.Query("SELECT * FROM MyTable").ToList();
        }
    }
}


I realise it doesn't save you much over the original code other than centralising the connection string...

A more advanced idea would be add an Execute method to your base repo but I don't think it would add much...

protected T Execute&lt;T&gt;(Func&lt;IDbConnection, T&gt; query)
{
    using (var connection = connectionFactory.GetOpenConnection())
    {
        return query(connection);
    }
}