Inclusion Criteria:
- Histologically confirmed adenocarcinoma of the prostate. Primary small cell carcinoma, carcinoid tumor, neuroendocrine (NE) carcinoma, or large cell NE carcinoma arising in the prostate are not allowed; however, adenocarcinomas with NE features (for example \[e.g.\], immunohistochemistry \[IHC\] with both androgen receptor \[AR\]- and NE-marker positivity) are allowed
- Must have metastatic castration-resistant prostate cancer (mCRPC)
- PSA must measure at least 2 nanograms per milliliters (ng/mL) at screening
- Measurable or evaluable disease
- Prior orchiectomy or medical castration; or, for participants who have not undergone orchiectomy, must be receiving ongoing androgen deprivation therapy with a gonadotropin-releasing hormone (GnRH) analog (agonist or antagonist) prior to the first dose of study drug and must continue this therapy throughout the treatment phase
- Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1
Exclusion criteria:
- Toxicity related to prior anticancer therapy that has not returned to grade less than or equal to (<=) 1 or baseline levels (except for alopecia and vitiligo)
- Known allergies, hypersensitivity, or intolerance to any of the components (e.g., excipients) of JNJ-78278343, JNJ-95298177, or JNJ-87189401
- Participants with leptomeningeal disease or brain metastases, with the exception of participants with definitively, locally treated brain metastases that are clinically stable and asymptomatic greater than (>) 2 weeks, and who are off corticosteroid treatment for at least 2 weeks prior to first dose of study treatment
- Treatment with any anti-cancer or investigational agents within 14 days prior to the first dose of study treatment; specific requirements for certain anti-cancer therapies are as follows:
1. Any T-cell redirecting treatment (e.g., CD3-directed bispecific or Chimeric Antigen Receptor T-cell \[CAR-T\] therapy) within 90 days prior to the first dose of study treatment
2. Immune checkpoint inhibitors within 6 weeks prior to the first dose of study treatment
3. Radium (Ra) 223 dichloride within 28 days prior to the first dose of study treatment
4. Any prior treatment with kallikrein-related peptidase 2 (KLK2)-targeted therapy
5. Any prior prostate-specific membrane antigen (PSMA)-targeting therapy (that is \[i.e.\], participants who received PSMA-targeting radioconjugates are excluded) \[Parts 2A and 2B only\]. Prior PSMA RLT is allowed in Part 1 and required for Part 2C and Part 2D but last dose must be >3 months prior to the first dose of study treatment
6. Any prior antibody drug conjugates (ADCs) with microtubule inhibitor payloads (e.g., auristatins, maytansinoids, tubulysins)
- Any serious underlying medical conditions or other issue that would impair the ability of the participant to receive or tolerate the planned treatment at the investigational site to understand the informed consent, or any condition for which, in the opinion of the investigator, participation would not be in the best interest of the participant or that could prevent, limit, or confound the protocol-specified assessments