Solution 1:

Sure, a function to create and dispose your Connection will work great.

protected void Execute(Action<IDbConnection> query)
{
    using (IDbConnection db = new SqlConnection(ConfigurationManager.ConnectionStrings["myDB"].ConnectionString))
    {
        query.Invoke(db);
    }
}

And your simplified call site:

public void SaveCustomer(CustomerDTO custDTO)
{
    Execute(db => db.Execute(saveCustSp, custDTO, CommandType.StoredProcedure));
}

With Return Values:

public T Get<T>(Func<IDbConnection, T> query)
{
    using (IDbConnection db = new SqlConnection(ConfigurationManager.ConnectionStrings["myDB"].ConnectionString))
    {
        return query.Invoke(db); 
    }
}

In your call site, just write the logic you wish to use.

public IEnumerable<EmployeeDTO> GetEmployeeDetails(int employeeId)
{
    return Get<IEnumerable<EmployeeDTO>(db => 
        db.Query<EmployeeDTO>(anotherSp, new { EmployeeID = employeeId }, CommandType.StoredProcedure));
}

Solution 2:

This is not directly relevant to your question. But I suggest you consider using DapperExtensions.

Initially, I did implemented Repository pattern using Dapper. The drawback was that, I have to write queries all over; it was very stringy. Due to hard coded queries it was near to impossible to write generic repository.

Recently, I upgraded my code to use DapperExtensions. This fixes lot many issues.

Following is the generic repository:

public abstract class BaseRepository<T> where T : BasePoco
{
    internal BaseRepository(IUnitOfWork unitOfWork)
    {
        dapperExtensionsProxy = new DapperExtensionsProxy(unitOfWork);
    }

    DapperExtensionsProxy dapperExtensionsProxy = null;

    protected bool Exists()
    {
        return (GetCount() == 0) ? false : true;
    }

    protected int GetCount()
    {
        var result = dapperExtensionsProxy.Count<T>(null);
        return result;
    }

    protected T GetById(Guid id)
    {
        var result = dapperExtensionsProxy.Get<T>(id);
        return result;
    }
    protected T GetById(string id)
    {
        var result = dapperExtensionsProxy.Get<T>(id);
        return result;
    }

    protected List<T> GetList()
    {
        var result = dapperExtensionsProxy.GetList<T>(null);
        return result.ToList();
    }

    protected void Insert(T poco)
    {
        var result = dapperExtensionsProxy.Insert(poco);
    }

    protected void Update(T poco)
    {
        var result = dapperExtensionsProxy.Update(poco);
    }

    protected void Delete(T poco)
    {
        var result = dapperExtensionsProxy.Delete(poco);
    }

    protected void DeleteById(Guid id)
    {
        T poco = (T)Activator.CreateInstance(typeof(T));
        poco.SetDbId(id);
        var result = dapperExtensionsProxy.Delete(poco);
    }
    protected void DeleteById(string id)
    {
        T poco = (T)Activator.CreateInstance(typeof(T));
        poco.SetDbId(id);
        var result = dapperExtensionsProxy.Delete(poco);
    }

    protected void DeleteAll()
    {
        var predicateGroup = new PredicateGroup { Operator = GroupOperator.And, Predicates = new List<IPredicate>() };
        var result = dapperExtensionsProxy.Delete<T>(predicateGroup);//Send empty predicateGroup to delete all records.
    }

As you can see in above code, most of the methods are just wrapper over underlying DapperExtensionsProxy class. DapperExtensionsProxy internally also manages UnitOfWork which you can see below. These two classes can be combined without any issue. I personally prefer to keep them separate.

You can also notice that additional methods Exists, DeleteById, and DeleteAll are implemented those are not part of DapperExtensionsProxy.

Method poco.SetDbId is defined in each POCO class to set its Identifier property. In my case, identifiers of POCOs may have different datatypes and names.

Following is DapperExtensionsProxy:

internal sealed class DapperExtensionsProxy
{
    internal DapperExtensionsProxy(IUnitOfWork unitOfWork)
    {
        this.unitOfWork = unitOfWork;
    }

    IUnitOfWork unitOfWork = null;

    internal int Count<T>(object predicate) where T : BasePoco
    {
        var result = unitOfWork.Connection.Count<T>(predicate, unitOfWork.Transaction);
        return result;
    }

    internal T Get<T>(object id) where T : BasePoco
    {
        var result = unitOfWork.Connection.Get<T>(id, unitOfWork.Transaction);
        return result;
    }

    internal IEnumerable<T> GetList<T>(object predicate, IList<ISort> sort = null, bool buffered = false) where T : BasePoco
    {
        var result = unitOfWork.Connection.GetList<T>(predicate, sort, unitOfWork.Transaction, null, buffered);
        return result;
    }

    internal IEnumerable<T> GetPage<T>(object predicate, int page, int resultsPerPage, IList<ISort> sort = null, bool buffered = false) where T : BasePoco
    {
        var result = unitOfWork.Connection.GetPage<T>(predicate, sort, page, resultsPerPage, unitOfWork.Transaction, null, buffered);
        return result;
    }

    internal dynamic Insert<T>(T poco) where T : BasePoco
    {
        var result = unitOfWork.Connection.Insert<T>(poco, unitOfWork.Transaction);
        return result;
    }

    internal void Insert<T>(IEnumerable<T> listPoco) where T : BasePoco
    {
        unitOfWork.Connection.Insert<T>(listPoco, unitOfWork.Transaction);
    }

    internal bool Update<T>(T poco) where T : BasePoco
    {
        var result = unitOfWork.Connection.Update<T>(poco, unitOfWork.Transaction);
        return result;
    }

    internal bool Delete<T>(T poco) where T : BasePoco
    {
        var result = unitOfWork.Connection.Delete<T>(poco, unitOfWork.Transaction);
        return result;
    }

    internal bool Delete<T>(object predicate) where T : BasePoco
    {
        var result = unitOfWork.Connection.Delete<T>(predicate, unitOfWork.Transaction);
        return result;
    }
}

Following is the BasePoco used above:

public abstract class BasePoco
{
    Guid pocoId = Guid.NewGuid();

    public Guid PocoId { get { return pocoId; } }

    public virtual void SetDbId(object id)
    {//Each POCO should override this method for specific implementation.
        throw new NotImplementedException("This method is not implemented by Poco.");
    }

    public override string ToString()
    {
        return PocoId + Environment.NewLine + base.ToString();
    }
}

This also uses UnitOfWork which is explained here.