This is a very interesting topic, and one that relates largely to my favorite genre.
I wouldn't be the first to say that shmups are ****ing hard. They are intense, brutal, and oftentimes utterly unforgiving, largely due to the nature of arcade games in general. And most arcade shmups of the eighties and nineties didn't have difficulty settings available to the player, though many home ports and console shmups did have the option. For arcade games, the difficulty of a game was solely in the hands of the arcade operator, and players either dealt with it or didn't play the games (or bought home versions, if those included difficulty settings). And players got to be too good at the games and started stretching their quarter out to the point that it made games unprofitable, so companies made new ones that were harder. And harder. And this absurd difficulty is what turns a lot of people away from shmups in particular.
But look at shmups nowadays. It's not uncommon for shmups, whether in arcades or at home, to offer a few difficulty options at the start. Some games go further, with the DeathSmiles games letting you pick your difficulty for each level (though you have to play each level on the hardest difficulty and on one credit to be able to go to the final stage). There's also stuff like the Touhou series, in which almost every game offers four distinct difficulty settings, and games like Space Invaders Extreme 2 and to a lesser extent Ikaruga, which offer "beginner" modes that give the player infinite lives (though in Ikaruga, this only applies to the first stage, and the whole game in this mode is only two stages long, and in Space Invaders Extreme 2, a player cannot choose their route or save a score if they play on beginner difficulty).
And with shmups in particular, I think giving the player a choice regarding difficulty is a good thing, because it means the genre isn't dying even as the games continue to get harder and harder. It means one player can play through a novice Ultra run on Mushihime-sama Futari and have plenty of fun, while another can play through a regular Ultra run and have just as much fun, if not more (if it's the sort of player who derives fun from overcoming challenges). It means the genre isn't being "dumbed down for the casuals" (an accusation thrown at many games in other, more popular genres).
As for the matter of "beating" a shmup, I tend to agree with most of my peers on the subject: when you have cleared the game on one credit at the default settings, you can say you have beaten the game. This isn't as simple as it might sound - for games such as Gradius, clearing the first "loop" would count as a clear, while for games such as DoDonPachi, defeating the true last boss is a clear (though some players will still say they've beaten a multi-loop-and-TLB game if they cleared the first loop on one credit, regardless of whether or not they met the requirements to enter the second loop). And "default settings" might vary as well, as with Battle Garegga, where "default settings" involve setting the region of the arcade version to Japan (the different regions have different starting rank and scores for extends), or the aforementioned Touhou games, where opinion is split between "Normal counts," "only Lunatic counts," "just make sure to note what difficulty you cleared it on when you're talking about it," and "Touhou sucks."
Now for everything else.
All games where difficulty is applicable should have difficulty settings. However, games with difficulty settings should also offer greater incentives for playing on higher difficulties, as in some of the shmup examples above. Another example would be The World Ends With You, where conducting battles on higher difficulties yields greater rewards; The World Ends With You also allows for more customization of the game's difficulty than pretty much any other game I've ever played.
Sometimes adjustable difficulty in-game works, if it's done in such a way that it makes the game accessible to more players without making it easy for everyone. That said, I agree with bobman that you should finish at the same difficulty where you started, unless you're increasing the difficulty because the game is too easy.
I also agree with everything bobman said regarding games that are too hard on their easiest difficulty being preferable to games that are too easy on their hardest difficulty, except for the part about super-easy games being super boring. Sometimes you play a game for the experience rather than the challenge; this is especially true today, since we now have all manner of "art" games.
I try to enjoy games where difficulty is a factor by way of optimization. Speed-running a Metroid game, earning the highest score I possibly can in Space Invaders Extreme 2 - these are things I enjoy. This isn't to say I don't enjoy games where difficulty isn't actually a factor the same way; for example, I try to clear JRPGs with the least amount of grinding possible, as opposed to just grinding and making the final battle end quickly. In most cases, the former is quicker.
Unlike Turtlekid, I absolutely refuse to use cheats, to the extent that I won't apply an Action Replay code to Yu-Gi-Oh! World Championship 2010: Reverse of Arcadia to give me every card even if it would unarguably increase my enjoyment of the game by allowing me to build any sort of deck I want. I'll instead earn more cards through playing the game or entering codes from the physical game cards to unlock them for purchase in the game, as was intended. However, I won't hesitate to abuse glitches in a game, such as the bug in Shao-Lin's Road that allows you to earn 1500 points from a single normal enemy. If a programming oversight in a game allows me to enjoy said game more or perform better, of course I'm going to take advantage of that.