B3LYP is still a decent functional at its level of theory (single-hybrid functional), but you're right that there's a general criticism of it, which I largely hear in the form of people saying things like "all they did was B3LYP/6-31G*" to criticize non-experts that blindly use this combination which became the "default" in chemistry for non-expert groups for many years.
Some history about B3LYP/6-31G*
As Susi Lehtola said here, BLYP/6-31G* performed better than wavefunction-based methods for equilibrium geometries, dipole moments, harmonic vibrational frequencies and atomization energies in this 1993 paper. A co-author of that paper (John Pople) was the original author of Gaussian70 and would soon get the Nobel Prize, so it was a widely seen paper, and then a year later Mike Frisch (currently President of Gaussian) did a comparison of BLYP to B3LYP and LSDA again with 6-31G* (and one more basis set), which led to the B3LYP/6-31G* combination becoming widely adopted.
But wavefunction-based methods have improved
In the above paragraph "wavefunction-based methods" referred only to MP2 and QCISD (it was 1993 after all) and if they could compare to CCSD(T) which became in the 2000s known as the "gold standard of quantum chemistry" for about a decade, they wouldn't have found found BLYP/6-31G* or B3LYP/6-31G* to perform better. Much of what people use B3LYP/6-31G* for can be done very fast with CCSD(T)/cc-pVTZ in 2021 if people just knew that, so many people find it cringeworthy when B3LYP/6-31G is still used.*
One should consider more than one functional
TAR86 pointed out here, that it's important to evaluate more than one functional for the set of molecules you're dealing with, or to look at benchmark studies which have already done this, and most of the time when you see people using B3LYP/6-31* they are using it with no justification apart from it being all they know. This means that people can again be getting worse results than what they could have got with the same cost, simply because they didn't do a bit more research.
Functionals have also improved (answering the call for "alternatives")
As the OP (Tristan Maxson) said here, B3LYP/6-31G* is old and there's now better options. As mentioned by me here, a comparison of 47 functionals (2 LDA, 14 GGA, 3 meta-GGA, 23 Hybrid, 5 Double Hybrid) in an enormous dataset was done by Goerigk and Grimme. You can see in the figure below that B3LYP is still a very decent functional, standing in the "middle" of the pack of other functionals at the same level of theory (single-hybrid), having an overall average error against CCSD(T)/experiment of slightly less than 4 kcal/mol while other single-hybrids can have average errors of a bit more than 2 kcal/mol in the best case and almost 5 kcal/mol in the worse case.
But you can indeed see several other functionals at the same level of theory of B3LYP:
So even if we want to blindly use just one functional as a black-box (without comparing against other functionals or looking for a benchmark study on the class of molecules we're dealing with), things like M06 and PW6B95 seem to work better better on average than B3LYP overall on a comprehensive dataset like the one used in the study above. They didn't use 6-31G* because some of the elements included in the dataset don't have 6-31G* sets available, but they did say that (aug-)def2-SV(P), which they did use, is usually comparable to 6-31G*.
Specific problems with B3LYP
The last section above answered your call for "alternatives at the same level of theory" so now I'd like to answer your call for:
"some of the more/less obvious problems that occur when using it"
The figure below, also from the same paper as the above figure, shows that B3LYP is:
- Quite good compared to other single-hybrids for basic properties (in this paper examples included atomization energies,
electron affinities, ionization potentials, proton affinities, SIE
related problems, barrier heights)
- A bit worse but still "in the middle of the pack" for non-covalent interactions (in this paper: water clusters, conformational energies, and inter- and intramolecular London-dispersion interactions)
- One of the worst overall for reaction energies (in this paper examples included isomerizations, Diels–Alder reactions, ozonolyses, reactions
involving alkaline metals)
So reaction energies would be one of the places to be careful about using B3LYP, in addition to RESP (Restrained Electrostatic Potential) calculations as noted in B. Kelly's warning that it will cause partial charges to be added implicitly (there's a lot more details in the original answer though), and for other things such as in Andrew Rosen's comment which I would be thrilled to see turned into an answer if time permits.