Chances are high that in addition to feeding 6V into a pin that's not even 5V tolerant, you're trying to run a 74HC part with woefully inadequate pin voltages.
The easiest solution is the proper voltage divider, run your sensor at 5V, and use a 74HCT165 (note the 'T' in the part designation). The 74HCT line is made to run from 5V and accept TTL input voltages -- which are, happily, about the same as what comes out of CMOS being driven from 3.3V.
If you absolutely must run your sensor at 6V, then you'll need proper voltage translation between it (or them) and your 74HCT165.
For transitions that are slower than about 100Hz or so (and maybe up to the low kHz), this should work well -- and you can collect Q1, R1 and R2 into a pre-biased transistor, for more savings.
- Choose R3 to be around 1k\$\Omega\$ to 10k\$\Omega\$
- Choose R1 and R2 so that if they were an unloaded voltage divider, they'd have a voltage of around 1.4 - 2V when the sensor output is on (i.e., \$\frac{R1}{R1 + R2} \simeq \frac{1.4\mathrm{V}}{V_{out}}\$). That uses the transistor's Vbe to make a really soft threshold of around half the sensor output voltage.
- Choose R1 and R2 so that their parallel equivalent is equal to about five times R3 or less (i.e., \$\frac{R1\,R2}{R1 + R2} \le 5 R3\$). That makes sure that the transistor is well into saturation when its on.
- Don't sweat getting the numbers for R1 and R2 exact.
simulate this circuit – Schematic created using CircuitLab