How to protect resources that may be used in a multi-threaded or async environment?

You can use SemaphoreSlim with 1 as the number of requests. SemaphoreSlim allows to lock in both an async fashion using WaitAsync and the old synchronous way:

await _semphore.WaitAsync()
try
{
    ... use shared resource.
}
finally
{
    _semphore.Release()
}

You can also write your own AsyncLock based on Stephen Toub's great post Building Async Coordination Primitives, Part 6: AsyncLock. I did it in my application and allowed for both synchronous and asynchronous locks on the same construct.

Usage:

// Async
using (await _asyncLock.LockAsync())
{
    ... use shared resource.
}

// Synchronous
using (_asyncLock.Lock())
{
    ... use shared resource.
}

Implementation:

class AsyncLock
{
    private readonly Task<IDisposable> _releaserTask;
    private readonly SemaphoreSlim _semaphore = new SemaphoreSlim(1, 1);
    private readonly IDisposable _releaser;

    public AsyncLock()
    {
        _releaser = new Releaser(_semaphore);
        _releaserTask = Task.FromResult(_releaser);
    }
    public IDisposable Lock()
    {
        _semaphore.Wait();
        return _releaser;
    }
    public Task<IDisposable> LockAsync()
    {
        var waitTask = _semaphore.WaitAsync();
        return waitTask.IsCompleted
            ? _releaserTask
            : waitTask.ContinueWith(
                (_, releaser) => (IDisposable) releaser,
                _releaser,
                CancellationToken.None,
                TaskContinuationOptions.ExecuteSynchronously,
                TaskScheduler.Default);
    }
    private class Releaser : IDisposable
    {
        private readonly SemaphoreSlim _semaphore;
        public Releaser(SemaphoreSlim semaphore)
        {
            _semaphore = semaphore;
        }
        public void Dispose()
        {
            _semaphore.Release();
        }
    }
}