I'm curious to see what the results are from Vox's asking people how they would change science. In my opinion there are lots of little things that could be better, but much of the problems faced by science today has more to do with funding and money than any other single item. In particular, I think that the common use of science departments as piggy banks for universities cuts off a lot of potential science, and hurts science education at those universities.
My solution is simple: better funding of public universities (general funds, not department specific). I submitted a [poorly organized] response to them to that end. The funding of universities (and science) is very complex, but the dramatic relative cuts in funding from states have led to the business style administrations that are more common than they used to be. Universities need a lot of money to operate. Some comes from student tuition and fees, some from private donations/endowments, some from grants and some (particularly for public/state schools) comes from government funding.
It used to be that quite a bit came from government funding (yes, I could probably find out how much but this is not a research paper), but the combination of decades of Republican tax cuts (which require subsequent spending cuts) compounded by the current lesser depression has shrunk that part down. Costs keep going up, however, and schools need to make up the difference. So tuition has risen...sharply, but that isn't enough. One of the places a university can find extra money is the grants that professors are awarded.
There are some different ways that universities get money from professor grants, but for every ten dollars of granted funds a university may get $3-5 of that. (Note: grants aren't all specified the same way. Sometimes a $1M grant means that the professor gets $500-700k, sometimes it means the professor gets $1M and the grant will pay the university whatever over that is necessary to make sure the professor gets the award total). Science grants are particularly nice ways to get money since they tend to be larger than grants in other departments/colleges. Some of this is because of the often expensive equipment necessary for science. Some of it is personnel, and some of it is, honestly, a value judgement. Most people consider science research more "wothwhile" compared to, say, humanities or fine arts research. (There is a bit of economics at play too, since there can be some financial rewards via technological advancement, but that often does not flow back to the grantor--particularly when the grantor is the government--it may, however, flow back to the university.)
The fact that science departments can be seen as profit centers for the university has some [potential] consequences. Most obvious is the publish or perish mentality, though that would probably be better reworded as "get funded or get out". While publications were once the hallmark of quality the number and quantity of grants fills that roll today. That isn't inherently bad, or it wouldn't be in a world where quality research could always find funding, but that isn't the world we live in.
Yet even accepting that there is another problem: established professors, have an easier time maintaining funding and getting new funding than those starting out. There are also "superstar" professors who have become funding juggernauts. This is easy to understand since bigger name professors tend to get higher quality students/post docs who do good work (sometimes developing that work) which further advances the name/opportunities of the professor. There are ebbs and flows and, again, this isn't by itself terrible since it can still lead to really good science, but the financial power these professors have at the universities has some downsides.
One is they often can negotiate contracts where they teach very little (sometimes not at all). Ok, yea, they are teaching their grad students/post docs but someone still needs to teach the undergrad classes/labs. That means hiring more profs, but they can also demand large salaries against a university that needs their money in part because they aren't funded well enough, and that can mean less cash for other professors, so now instead of hiring a tenure-track position, they are hiring adjuncts: basically glorified post docs that get minimal salary and poor to no benefits. PhD's teaching science as adjuncts at universities can be compensated more poorly than our under-compensated high school teachers.
Another issue with these super-profs is that their talents (mostly the one for getting lots of grant money to roll in) are in high demand. The number of senior positions available at universities is larger than ever, while the number of entry level tenure-track positions is dwindling. So while well funded big name professors can go pretty much wherever they want, aspiring researchers have to fight a lot harder just to get in the door. (Note: large private research institutions still do hire a fair number of entry levels, but getting that job a Harvard, which has always been difficult, is even harder today since there are fewer StateU positions to go around.)
I should add in: super-profs are great. They nurture talent, their labs can be sources for scientific breakthroughs, they can be good for public outreach (more of a mixed bag here but still). Were it not for universities being desperate to get more money these profs would still exist, but the universities they work at would not be so beholden to them, and more people would have the chance to find their way to that roll. Also, maybe the students would be getting better education.