There are roughly two approaches that the Democrats could take to Trump's presidency. One would be to start attacking him before the inauguration, question the legitimacy of his presidency, and seek to thwart him at every turn. This has the advantage of keeping him on the defensive and might restrict the amount of things he can get done in the form of legislation and nominations. If he ever really screws up, the Democrats will be in a position to say "We told you so" and take advantage of the situation to win the mid-term elections, the following presidency, and maybe even get him impeached.
One disadvantage to this approach is that it gives the Democrats little extra leverage if Trump does something outrageous but not explicitly illegal. With their complaints already turned up to maximum volume, they won't be able to distinguish an outrageous act from any other kind (much as they were unable to paint Trump as drastically different from other Republican presidential candidates) and people who haven't already made their minds up might well shrug and ignore them.
The other disadvantage is that it is utterly hypocritical. President Obama insisted a few months ago that there was no way the election could be hacked, and various Democrats declared that it would be almost treasonous to question the validity of our electoral process. Now, of course, they insist that the Russians did somehow hack the election and that Trump's victory is therefore tainted. I don't know which is worse, the plain ignorance of the claim or the fact that no Democrat that I have seen has owned up to the contradiction.
It is also hypocritical to start opposing Trump before he has done anything. These are the same Democrats who declared Republicans "obstructionists" for trying to block Obama's every move as soon as he became president. Obama won, therefore Republicans needed to "work with" him instead of working against him. Instead of modeling correct behaviour, they are now one-upping Republicans by opposing Trump before he has even done anything. Every nomination, every speech, every alteration of the White House website is a new sign of Trump's degeneracy and unfitness to rule.
Self-aware Democrats are in short supply these days: they definitely exist, but they are a small minority in the party. I don't deny people the right to change their minds. I'm all for it; I have been known to do it myself sometimes, even on fundamental matters. The thing is, when someone is wrong to the point of directly contradicting something he has said previously, you might expect him to move forward with more humility and consciousness of man's propensity to error than he did before. Not so with these Democrats. They are confident with a capital "C" that Trump is going to be a bad president, so confident that they have already condemned him before he was even sworn into office.
(I like to imagine what would happen if Trump, being the person he is, changed his political beliefs while in office and won the presidence in 2020 as a Democrat. I'm quite sure we would then get the same chorus of "respect the dignity of the office" and "work with the president" that we heard in 2009 and 2013.)
So what is the alternative? How about Democrats at least set a good example and declare their willingness to work with President Trump to move the nation forward. They don't have to agree to anything he proposes, but at least they could disagree in good faith and with a smile. Sure, it might not get the same short-term results as yelling and screaming. However, if Trump ever crosses the line (and who really doubts that he will at some point?), Democrats would then be in a position to say, "We gave you a chance, we tried to work with you, but now you are violating the Constitution and we have to take all due measures to stop you," including impeachment. They would have so much more credibility in that circumstance. As it is, I think anyone who isn't already committed to the progressive cause is going to say "ho-hum" and ignore the Democrats complaints. The only way this could work out for them is if Trump does something so plainly wrong that everyone has no choice but to admit it, something like using a government agency to suppress his opponents. That is possible, but these things are rarely so clear-cut. More likely, Trump will push the boundaries here and there and generally make the Democrats uncomfortable, without violating any laws. And their strategy is very poorly adapted to handle that.