Based on the answer at https://stackoverflow.com/questions/6787318/set-alsa-master-volume-from-c-code/6787957#6787957 and the native interop code from alsa-sharp (https://github.com/atsushieno/alsa-sharp/) by @atsushieno, the code in this gist presents a solution for "how to set the volume on Linux in C# through Alsa" that works for me.

AlsaVolumeControl.cs

using System;
using System.Runtime.InteropServices;

namespace Alsa
{
    public sealed class VolumeControl : IDisposable
    {
        private readonly long _min;
        private readonly long _max;
        private readonly IntPtr _sid;
        private readonly IntPtr _selem;
        private readonly IntPtr _handle;
        private const string LibraryName = "libasound";

        public VolumeControl(string card = "default", string selemName = "PCM")
        {
            snd_mixer_open(ref _handle, 0);
            snd_mixer_attach(_handle, card);
            snd_mixer_selem_register(_handle, default, default);
            snd_mixer_load(_handle);

            snd_mixer_selem_id_malloc(ref _sid);
            snd_mixer_selem_id_set_index(_sid, 0);
            snd_mixer_selem_id_set_name(_sid, selemName);
            _selem = snd_mixer_find_selem(_handle, _sid);

            snd_mixer_selem_get_playback_volume_range(_selem, ref _min, ref _max);
        }

        public void SetVolumePercent(int volume)
        {
            snd_mixer_selem_set_playback_volume_all(_selem, (int)(volume * _max / 100));
        }

        public void Dispose()
        {
            ReleaseUnmanagedResources();
            GC.SuppressFinalize(this);
        }

        private void ReleaseUnmanagedResources()
        {
            snd_mixer_selem_id_free(_sid);
            snd_mixer_close(_handle);
        }

        ~AlsaSoundOutput()
        {
            ReleaseUnmanagedResources();
        }

        [DllImport(LibraryName)]
        internal static extern int snd_mixer_open(ref IntPtr mixer, int mode);

        [DllImport(LibraryName, CharSet = CharSet.Ansi)]
        internal static extern int snd_mixer_attach(IntPtr mixer, [MarshalAs(UnmanagedType.LPStr)] string name);

        [DllImport(LibraryName)]
        internal static extern int snd_mixer_selem_register(IntPtr mixer, IntPtr options, IntPtr classp);

        [DllImport(LibraryName)]
        internal static extern int snd_mixer_load(IntPtr mixer);

        [DllImport(LibraryName)]
        internal static extern int snd_mixer_selem_id_malloc(ref IntPtr selem);

        [DllImport(LibraryName)]
        internal static extern void snd_mixer_selem_id_set_index(IntPtr selem, uint val);

        [DllImport(LibraryName, CharSet = CharSet.Ansi)]
        internal static extern void snd_mixer_selem_id_set_name(IntPtr selem, [MarshalAs(UnmanagedType.LPStr)] string value);

        [DllImport(LibraryName)]
        internal static extern IntPtr snd_mixer_find_selem(IntPtr mixer, IntPtr selem);

        [DllImport(LibraryName)]
        internal static extern int snd_mixer_selem_get_playback_volume_range(IntPtr selem, ref long min, ref long max);

        [DllImport(LibraryName)]
        internal static extern int snd_mixer_selem_set_playback_volume_all(IntPtr selem, int value);

        [DllImport(LibraryName)]
        internal static extern void snd_mixer_selem_id_free(IntPtr selem);

        [DllImport(LibraryName)]
        internal static extern int snd_mixer_close(IntPtr mixer);
    }
}