If there is an inherent problem with the Muddler Minnow it is that the large, buoyant head makes it difficult to get the fly to sink. And for a fly meant to imitate sculpins, a benthic fish without a swim bladder, it needs to get down to be fished as a sculpin imitation. Of course, fairly often, the fly is not imitating a sculpin but some other small fish.
There is an issue with weighting flies, it is that they start to behave differently in the water. They get "jiggier" or they sink somewhat unnaturally. So before we get to weighting flies, there are options for fishing unweighted flies. First is how the fly is tied. If you want a fly that gets down without weight on the fly or the leader, tie the head more sparsely. Large, buoyant heads have their place but they tend to fish better on sinking lines or leaders. One of the ways to get your fly down is to shorten up your leader. Fishes are generally much less leader shy when it comes to streamers. Fishing a buoyant fly on a long leader makes the fly harder to sink.
Lastly, and maybe most significantly, using the force of water to sink your flies and manipulating your line to help the fly sink faster. While many go to weighted flies, lines, or leaders first; line manipulation can get your fly down with a lot less disturbance. Using mends can help get an unweighted fly down to where you want it.
Probably the most difficult but maybe more effective way to get your fly to sink is to use water currents to your advantage. This typically means fishing from above the fish and fishing down and across. Maybe first, a mending primer is needed - go read the MidCurrent article...or watch the video below.
In general, mending upstream will slow the fly's movement across the stream and mending downstream will speed up its movement across the stream which generally causes it to rise in the water column. However on a sinking line or leader you can counteract this by using water pressure to push against the line (but it is not easy to do).
George Daniel - How to Fish Streamers in Riffles (Fly Fisherman magazine)
Chad Shmukler - 5 Tips for Better Streamer Fishing (Hatch Magazine)
Clint Losee - Streamer Fishing Tactics (The Fly Fishing Basics)
Weighting the Line and / or Leader
There are no shortage of options for weighting the line or the leader - I present them here from the simple to the more evolved and specialized. Sometimes the simple answer is the best answer.
Adding weight - like split shot, lead strips, or a heavy putty - to the leader is the simplest way to get your fly a little deeper. This does not always make casting fun nor water entry subtle but it does work and is probably the simplest and least involved method to fish a deeper spot or two. And the angler has some control over how far above the bottom the fly is fished by how far above the fly the weight is placed. Additionally, placing the weight closer to the hook makes the fly "jiggier" and placing it further away typically allows the fly to more more erratically.
Sinking leaders give you a great number of length and sink rate options and are the method I uses most of the time - I'll explain why. They are quick, simple, and easy to use. As a very part time streamer angler, I do not have a desire to carry around spare spools rigged with sink tips or full sink lines, particularly when wading. Taking off my monofilament leader and replacing it with a sinking poly leader is a simple way to achieve a similar(ish) result. I can find sink leaders is 7 and 12 foot lengths and in different inch per second sink rates to change how the leader, and thus the fly, sinks. This gives me enough options for Driftless streams - you might want more on larger water where you are casting much more than 30 feet at a time.
Sinking lines are not just the domain of lake anglers. They can be a little hard to handle, particularly on smaller streams but on larger rivers, an intermediate or full sink line can be a great asset. Increasingly there are sinking lines made for moving water - this is particularly true at the larger line weights. Finding a sinking line below a 6 weight can be more difficult.
How deeply the fly is fished can be controlled by a number of factors. Of course the amount of weight is the primary determinant of how deeply you can get your fly. However, couple weight with mending and you can have greater control of the depth.
Weighting the Fly
Again, there are a ton of options for how to weight a fly to help get it down. And these options range from the easy and more subtle to a generally heavier and less natural presentation.
The hook is the first line in weighting a fly. With streamers, you have a pretty great amount of choices for hook weights. Having played around weighing some hooks, there is a great amount of variation in their weights. Bass "stinger hooks" are a favorite streamer hook of mine and they can drastically change a fly pattern. A #2 Gamakatsu B10S averages about 305 milligrams (4.7 grains), the Tiemco equivalent - TMC8089 - is half again heavier at 454 milligrams (7.0 grains), and the Mustad C52SNP-BN is quite similar to the Gamakatsu hook at 306 milligrams. This can make a huge difference in how the fly sinks and behaves.
Non-visible weight typically means adding lead or non-toxic wire to the shank and covering it with the body material. There are also some flies that hide beads or cones internally. Compared to hooks, this is a heavier or at least more concentrated option. Hook weight is spread over the entire hook, weights like beads, cones, or wire are much more densely concentrated.
Visible weight like beads, cones, dumbbell eyes, and formed heads are other options. Increasingly, adding UV epoxy is a way to weight a fly. The Clouser Deep Minnow is probably the best known option. For muddlers - a slider is basically a muddler with inverted dumbbell eyes. And beads and particularly cones are typical additions to a mudder head. For example, the rolled muddler often has a bead at its front and the conehead muddler makes for a quite an attractive fly.
Putting it Together
I find putting overly heavy flies on sinking lines or leaders is not a ton of fun to deal with and the flies that work best on sinking lines / leaders have some buoyancy to them. The large head of a muddler is a good water pusher that allows trout (and other fish) to feel the fly. I am mostly fishing these technique in the early season and again by late in the season as the bugs - both aquatic and terrestrial - are not terribly active.
And again in all honesty, I am mostly a dry fly snob and this is a pretty minor technique for me but above are plenty of videos and plenty of links below for this to serve as a place to learn more. Like anything fly fishing, you have to get out there and try it.