A radio transmitter can definitely induce a current in wires. That’s precisely what radio transmitters do.
According to this PDF (see page 2), blasting caps are designed to have a minimum firing current of 0.25 amps. According to this discussion, blasting cap resistance is on the order of 1-3 ohms, so if power is I2R, then your phone or FRS radio needs to deliver 125 milliwatts of power to the wires connected to the blasting cap. Cell phones transmit with 3 watts at most - in other words, you’d need those blasting cap wires to receive 4% of your phone’s transmitted power. If you’re in a place where the general public is allowed (e.g. on a road passing by a blasting zone), this seems highly unlikely to happen, especially if the blasting crew has done taken the appropriate steps to minimize the blasting circuits susceptibility to radio frequency interference (see guide two paragraphs below).
Having said that, this article reports that premature detonations have happened when blasting crew team members were using 2-way radios, but these were folks who were working on site, likely much closer to the explosives/wires than the general public can get. No further details were provided, but the cited expert appears to eat/sleep/breathe explosives, so you can probably take her at her word.
For further reading, there’s this PDF, “A Guide to Radio Frequency Hazards With Electric Detonators.” Turns out AM radio transmitter antennas are the biggest problem because of their low frequency and high power. Cell phones and mobile radios us much higher frequencyes, but they are still considered a problem specifically because they can be brought into the blasting area. Tables in section 2 (page 11) list minimum recommended safe distances for a variety of radio sources. Table 6 (page 15) shows that for “public use cellular telephones above 800 MHz”, the minimum safe distance is somewhere between 8 and 18 feet.
Page 6 begins a review of radio pickup circuits, i.e. configurations of the wiring in a blasting circuit that make them more likely to pick up RFI. The most common hazards appear to be having wires with a length around 1/4 of the radio wavelength, and/or having the wires in a circuit separated from each other and/or the ground. It’s surprising to me that they don’t use twisted-pair wires, which are a common thing in computer wiring to minimize susceptibility to induced currents.
Good blasting safety practice would mean careful construction of the blasting circuits to minimize RFI susceptibility, safe storage of blasting caps in Faraday-cage packaging, consideration of known nearby transmitting antennas, rigorous exclusion of mobile transmitters from the blasting site, and keeping the general public (and any transmitting devices they may have) far away. In the end, it seems like telling passing motorists to turn off their phones is just one more low-cost slice of the swiss cheese safety model. Those signs will never get everybody to turn their phones off, but the compliance of even a few is helpful.