As far as I am aware, no studies have been specifically conducted on the Van Allen belts' effect on humans. However, I reject the notion that such a specific study is necessary to determine safe exposure limits and travel times. Safety guidelines and regulations already exist for exposure to the same kinds of radiation here on earth. It is standard practice in nuclear and radiological fields to monitor personnel radiation dosage through the use of portable dosimetry equipment, to ensure that accumulated exposure over a given period of time does not exceed very conservative limits set by the Nuclear Regulatory Commission (see here).
I went looking for good data on measured proton and electron flux densities in the Van Allen belts, but there are too many different data points for various energy levels floating around online for me to want to deal with. Since all I care about is biological damage, I decided to narrow my scope to absorbed radiation dose measurements taken by Nasa's Van Allen probes, using conservative values in my calculations. Nasa's Van Allen probes A & B, launched in 2012, collected cumulative absorbed dose readings over a 971 day period. Exposure Radiation Monitor B recorded a value of 9660 rad over that period with aluminum shielding 0.9 cm thick (see Table 2 of this document). At greater than or equal to 3 mm depth of aluminum shielding, the protons of the inner Van Allen belt dominated the observed cumulative dose. At 1.5 mm or less of aluminum shielding, the electrons of the outer belt dominated the observed cumulative dose. This was because electrons, having less mass but equal electric charge to a proton, are more easily absorbed or scattered by aluminum shielding than the protons. Thus, electrons were only observable on dosimetry equipment at aluminum thicknesses less than 3 mm.
The unit for absorbed dose "rad" (Radiation Absorbed Dose) is a measure of energy imparted into a material by absorbed radiation (see this website). The absorbed dose equivalent unit "rem" is used to measure biological damage caused by a given kind of radiation (see previous link). Since biological damage is what I care about, I converted the measured 9660 rad to rem by multiplying by a Quality Factor (QF, relates energy imparted to damage caused by radiation type). I know the Van Allen belts - collectively - are dominated by electron flux, with a lower proportion of particles being protons. So, I chose a conservative value of 5 for my quality factor (accounting for electrons and protons, given their time- and location-variable flux densities). Electrons are consistently listed in references as having a QF of 1. However, protons have been listed in older references with a value of ~10 QF, while more recent references have adjusted it to ~5 or even as low as ~2. (What this means is that a proton, while less common than an electron, causes anywhere from 2x to 5x the biological damage). I consider a QF of 5 to be higher than the expected value for the combined electron/proton flux, and thus a suitable conservative value. See Radiation Weighting Factors.
Running through the math, (9660 rad) x (5 QF) = 48300 rem. (48300 rem) / (971 days) = 49.7 rem/day, or 2.07 rem/hr given 0.9 cm of aluminum shielding.
To calculate the effective aluminum shielding thickness of a spacecraft, I used the Apollo Command Module as an example (using this as a reference). The "hull" of the Apollo Command Module was actually very thin, averaging around a 4 mm thick sheet of aluminum. However, in radiation shielding analysis (at least in this case, where radiation lacks notable directionality), all structural elements of the craft need to be accounted for, not just the hull. Using some complicated calculus that I am unable and unwilling to do, but the author of the aforementioned text was kind enough to calculate for me: "[an] average of the protection provided by Apollo6 is assumed excluding the ablative region (already accounted for by the propulsion module). This number is 6.15 g/cm2 (12.6 lb/ft2) of aluminum." Knowing that the density of aluminum is approximately 2.70 g/cm^3, I can calculate the effective thickness of that shielding by dividing (6.15 g/cm^2) / (2.70 g/cm^3) = ~2.27 cm. (EDIT: I just found a source claiming they had 25 mm of aluminum shielding... so my number might be more conservative than I thought: link).
Next, I extrapolated the tenth thickness of aluminum for protons from the Van Allen probe's measured exposure at 3 mm and 9 mm thickness. I used the equation found here to do this. This equation is specific to gamma radiation, but works well enough for rough approximations of other kinds of radiation, too. Rearranging and plugging in data from the Van Allen probe, I calculated that about 1.5 cm of aluminum was sufficient to lower radiation exposure due to protons by a factor of ten. Plugging into the equation, again, this time using an "intensity" of 2.07 rem/hr (the equation works with rem/hr just as well as rad - it's a ratio) and an additional thickness of (2.27 cm - 0.9 cm = 1.37 cm) resulted in a value of approximately 0.26 rem/hr.
So, I'd estimated that the Apollo astronauts received an absorbed dose equivalent of 0.26 rem/hr while passing through the Van Allen belts. Given that the Apollo astronauts only spent a matter of hours crossing the belts (6 according to one of my aforementioned sources - 3.5 out, 2.5 on the way back to earth), I expect they would have received about 1.6 rem in total from the Van Allen belts. To see if any safety limits would have been exceeded, I referenced the Department of Energy Radiological Control Standard ([DOE-STD-1098-2008](www.energy.gov/sites/prod/files/2018/02/f48/DOE 2008_STD-1098 Rad Control.pdf)). The DoE whole body radiation dose limit for general employees is 5 rem/yr (though individual facilities impose administrative limits less than that). That limit is for a long-term, chronic dose, which risks stochastic effects such as cancer. However, in an acute dose (high radiation in a short time period), 25 rem is sufficient to cause temporary sterility in men, 100 rem may cause nauseau or skin reddening, and 500 rem is the generally accepted amount which will kill a person (link).
1.6 rem is well below the dosage needed to exceed the long term limit of 5 rem/yr, and not even close to the minimum needed for acute radiation poisoning (25 rem). By my admittedly basic calculations, they only received the equivalent of ~5 years worth of the exposure everyone on earth receives from natural sources (see DOE-HDBK-1131-98). Please also keep in mind that modern spacecraft utilize more effective shielding materials and methodologies than what was used during the Apollo program, further reducing the risk.